Engineering Mechanics · Dynamics · Open Problem Companion

Chapter Twelve
Kinematics of a Particle

12.1 Rectilinear Kinematics ~80 problems
Preliminary
  1. P12‑1aIf \(s = (2t^3)\) m, where \(t\) is in seconds, determine \(v\) when \(t = 2\) s.
  2. P12‑1bIf \(v = (5s)\) m/s, where \(s\) is in meters, determine \(a\) at \(s = 1\) m.
  3. P12‑1cIf \(v = (4t + 5)\) m/s, where \(t\) is in seconds, determine \(a\) when \(t = 2\) s.
  4. P12‑1dIf \(a = 2\) m/s², determine \(v\) when \(t = 2\) s if \(v = 0\) when \(t = 0\).
  5. P12‑1eIf \(a = (s)\) m/s², where \(s\) is in meters, determine \(v\) when \(s = 5\) m if \(v = 0\) at \(s = 4\) m.
  6. P12‑1fIf \(a = (s)\) m/s², where \(s\) is in meters, determine \(v\) when \(s = 5\) m if \(v = 0\) at \(s = 4\) m.
  7. P12‑1gIf \(a = 4\) m/s², determine \(s\) when \(t = 3\) s if \(v = 2\) m/s and \(s = 2\) m when \(t = 0\).
  8. P12‑1hIf \(a = (8t^2)\) m/s², determine \(v\) when \(t = 1\) s if \(v = 0\) at \(t = 0\).
  9. P12‑1iIf \(s = (3t^2 + 2)\) m, determine \(v\) when \(t = 2\) s.
  10. P12‑1jWhen \(t = 0\) the particle is at \(A\). In four seconds it travels to \(B\), then in another six seconds it travels to \(C\). Determine the average velocity and the average speed. The origin is at \(O\). figure — add your own
Fundamental
  1. F12‑1Initially, the car travels along a straight road with a speed of \(35\) m/s. If the brakes are applied and the speed is reduced to \(10\) m/s in \(15\) s, determine the constant deceleration. figure — add your own
  2. F12‑2A ball is thrown vertically upward with a speed of \(15\) m/s. Determine the time of flight when it returns to its original position. figure — add your own
  3. F12‑3A particle travels along a straight line with \(v = (4t - 3t^2)\) m/s, where \(t\) is in seconds. Determine the position when \(t = 4\) s. \(s = 0\) when \(t = 0\).
  4. F12‑4A particle travels along a straight line with \(v = (0.5t^3 - 8t)\) m/s, where \(t\) is in seconds. Determine the acceleration when \(t = 2\) s.
  5. F12‑5The position is \(s = (2t^2 - 8t + 6)\) m, where \(t\) is in seconds. Determine the time when \(v = 0\), and the total distance traveled when \(t = 3\) s. figure — add your own
  6. F12‑6A particle travels along a straight line with \(a = (10 - 0.2s)\) m/s², where \(s\) is in meters. Determine \(v\) when \(s = 10\) m if \(v = 5\) m/s at \(s = 0\). figure — add your own
  7. F12‑7A particle moves along a straight line with \(a = (4t^2 - 2)\) m/s², where \(t\) is in seconds. When \(t = 0\), the particle is \(2\) m to the left of the origin, and when \(t = 2\) s it is \(20\) m to the left. Determine the position when \(t = 4\) s.
  8. F12‑8A particle travels along a straight line with \(v = (20 - 0.05s^2)\) m/s, where \(s\) is in meters. Determine the acceleration at \(s = 15\) m.
Problems
  1. 12‑1Starting from rest, a particle moving in a straight line has \(a = (2t - 6)\) m/s², where \(t\) is in seconds. What is the velocity when \(t = 6\) s, and what is its position when \(t = 11\) s?
  2. 12‑2The acceleration of a particle as it moves along a straight line is \(a = (4t^3 - 1)\) m/s², where \(t\) is in seconds. If \(s = 2\) m and \(v = 5\) m/s when \(t = 0\), determine the particle's velocity when \(t = 5\) s. Also determine the total distance traveled during this time period.
  3. 12‑3The velocity of a particle traveling in a straight line is \(v = (6t - 3t^2)\) m/s, where \(t\) is in seconds. If \(s = 0\) when \(t = 0\), determine the particle's deceleration and position when \(t = 3\) s. How far has the particle traveled during the 3‑s interval, and what is its average speed?
  4. 12‑4A particle is moving along a straight line such that its position is \(s = (10t^2 + 20)\) mm, where \(t\) is in seconds. Determine (a) the displacement from \(t = 1\) s to \(t = 5\) s, (b) the average velocity during this interval, and (c) the acceleration when \(t = 1\) s.
  5. 12‑5A particle moves along a straight line such that its position is \(s = (t^2 - 6t + 5)\) m. Determine the average velocity, the average speed, and the acceleration when \(t = 6\) s.
  6. 12‑6A stone \(A\) is dropped from rest down a well, and \(1\) s later another stone \(B\) is dropped from rest. Determine the distance between the stones another second later.
  7. 12‑7A bus starts from rest with a constant acceleration of \(1\) m/s². Determine the time required to attain a speed of \(25\) m/s and the distance traveled.
  8. 12‑8A particle travels along a straight line with \(v = (12 - 3t^2)\) m/s, where \(t\) is in seconds. When \(t = 1\) s, the particle is \(10\) m to the left of the origin. Determine the acceleration when \(t = 4\) s, the displacement from \(t = 0\) to \(t = 10\) s, and the distance traveled during this time.
  9. 12‑9When two cars \(A\) and \(B\) are next to one another, they are traveling in the same direction with speeds \(v_A\) and \(v_B\) respectively. If \(B\) maintains its constant speed while \(A\) begins to decelerate at \(a_A\), determine the distance \(d\) between the cars at the instant \(A\) stops. figure — add your own
  10. 12‑10A particle travels along a straight‑line path such that in \(4\) s it moves from \(s_A = -8\) m to \(s_B = +3\) m. Then in another \(5\) s it moves from \(s_B\) to \(s_C = -6\) m. Determine the particle's average velocity and average speed during the \(9\)‑s interval.
  11. 12‑11Traveling with an initial speed of \(70\) km/h, a car accelerates at \(6000\) km/h² along a straight road. How long will it take to reach \(120\) km/h? Through what distance does the car travel during this time?
  12. 12‑12A particle moves along a straight line with \(a = 5/(3s^{1/3} + s^{5/2})\) m/s², where \(s\) is in meters. Determine the velocity when \(s = 2\) m if it starts from rest when \(s = 1\) m. Use a numerical method to evaluate the integral.
  13. 12‑13The acceleration of a particle as it moves along a straight line is \(a = (2t - 1)\) m/s², where \(t\) is in seconds. If \(s = 1\) m and \(v = 2\) m/s when \(t = 0\), determine the particle's velocity and position when \(t = 6\) s. Also determine the total distance traveled during this time.
  14. 12‑14A train starts from rest at station \(A\) and accelerates at \(0.5\) m/s² for \(60\) s. Afterwards it travels with constant velocity for \(15\) min. It then decelerates at \(1\) m/s² until brought to rest at station \(B\). Determine the distance between the stations.
  15. 12‑15A particle is moving along a straight line such that its velocity is \(v = (-4s^2)\) m/s, where \(s\) is in meters. If \(s = 2\) m when \(t = 0\), determine the velocity and acceleration as functions of time.
  16. 12‑16Determine the time required for a car to travel \(1\) km along a road if the car starts from rest, reaches a maximum speed at some intermediate point, and then stops at the end. The car can accelerate at \(1.5\) m/s² and decelerate at \(2\) m/s².
  17. 12‑17A particle is moving with velocity \(v_0\) when \(s = 0\) and \(t = 0\). If it is subjected to a deceleration \(a = -kv^3\), where \(k\) is constant, determine its velocity and position as functions of time.
  18. 12‑18A particle is moving along a straight line with an initial velocity of \(6\) m/s when it is subjected to a deceleration \(a = (-1.5v^{1/2})\) m/s², where \(v\) is in m/s. Determine how far it travels before it stops. How much time does this take?
  19. 12‑19The acceleration of a rocket traveling upward is \(a = (6 + 0.02s)\) m/s², where \(s\) is in meters. Determine the rocket's velocity when \(s = 2\) km and the time needed to reach this altitude. Initially \(v = 0\) and \(s = 0\) when \(t = 0\). figure — add your own
  20. 12‑20The acceleration of a rocket traveling upward is \(a = (6 + 0.02s)\) m/s², where \(s\) is in meters. Determine the time needed for the rocket to reach an altitude of \(s = 100\) m. Initially \(v = 0\) and \(s = 0\) when \(t = 0\). figure — add your own
  21. 12‑21When a train is traveling along a straight track at \(2\) m/s, it begins to accelerate at \(a = (60v^{-4})\) m/s², where \(v\) is in m/s. Determine its velocity and position \(3\) s after the acceleration begins. figure — add your own
  22. 12‑22The acceleration of a particle along a straight line is \(a = (2t - 9)\) m/s², where \(t\) is in seconds. At \(t = 0\), \(s = 1\) m and \(v = 10\) m/s. When \(t = 9\) s determine (a) the position, (b) the total distance traveled, and (c) the velocity.
  23. 12‑23If atmospheric resistance is accounted for, a freely falling body has \(a = 9.81[1 - v^2(10^{-4})]\) m/s², where \(v\) is in m/s and positive downward. If released from rest at a very high altitude, determine (a) the velocity when \(t = 5\) s and (b) the terminal velocity as \(t \to \infty\).
  24. 12‑24A sandbag is dropped from a balloon ascending vertically at \(6\) m/s. If the bag is released with the same upward velocity of \(6\) m/s when \(t = 0\) and hits the ground when \(t = 8\) s, determine the speed as it hits the ground and the altitude of the balloon at this instant.
  25. 12‑25A particle is moving along a straight line such that its acceleration is \(a = (-2v)\) m/s², where \(v\) is in m/s. If \(v = 20\) m/s when \(s = 0\) and \(t = 0\), determine the particle's position, velocity, and acceleration as functions of time.
  26. 12‑26The acceleration of a particle traveling along a straight line is \(a = \frac{1}{4}s^{1/2}\) m/s², where \(s\) is in meters. If \(v = 0\), \(s = 1\) m when \(t = 0\), determine the velocity at \(s = 2\) m.
  27. 12‑27When a particle falls through air, its initial acceleration \(a = g\) diminishes until it is zero, and thereafter it falls at terminal velocity \(v_f\). If \(a = (g/v_f^2)(v_f^2 - v^2)\), determine the time needed for the velocity to become \(v = v_f/2\). Initially the particle falls from rest.
  28. 12‑28A sphere is fired downward into a medium with an initial speed of \(27\) m/s. If it experiences a deceleration \(a = (-6t)\) m/s², where \(t\) is in seconds, determine the distance traveled before it stops.
  29. 12‑29A ball \(A\) is thrown vertically upward from the top of a \(30\)-m-high building with an initial velocity of \(5\) m/s. At the same instant another ball \(B\) is thrown upward from the ground with \(20\) m/s. Determine the height from the ground and the time at which they pass.
  30. 12‑30A boy throws a ball straight up from the top of a \(12\)-m-high tower. If the ball falls past him \(0.75\) s later, determine the velocity at which it was thrown, the velocity when it strikes the ground, and the time of flight.
  31. 12‑31The velocity of a particle traveling along a straight line is \(v = v_0 - ks\), where \(k\) is constant. If \(s = 0\) when \(t = 0\), determine the position and acceleration as a function of time.
  32. 12‑32Ball \(A\) is thrown vertically upward with velocity \(v_0\). Ball \(B\) is thrown upward from the same point with the same velocity \(t\) seconds later. Determine the elapsed time \(t < 2v_0/g\) from the instant ball \(A\) is thrown to when the balls pass each other, and find the velocity of each ball at this instant.
  33. 12‑33As a body is projected to a high altitude, the variation of gravitational acceleration with altitude \(y\) is \(a = -g_0[R^2/(R+y)^2]\), where \(g_0 = 9.81\) m/s² and \(R = 6356\) km. Determine the minimum initial velocity (escape velocity) at which a projectile should be shot vertically from the earth's surface so that it does not fall back. figure — add your own
  34. 12‑34Accounting for the variation of gravitational acceleration with altitude \(y\), derive an equation that relates the velocity of a freely falling particle to its altitude. Assume released from rest at altitude \(y_0\). With what velocity does it strike the earth if released from \(y_0 = 500\) km? Use the numerical data in Prob. 12‑33.
12.2 Erratic Motion ~50 problems
Preliminary
  1. P12‑2aDraw the \(s\)–\(t\) and \(a\)–\(t\) graphs if \(s = 0\) when \(t = 0\). figure — add your own
  2. P12‑2bDraw the \(a\)–\(t\) and \(v\)–\(t\) graphs. figure — add your own
  3. P12‑2cDraw the \(v\)–\(t\) and \(s\)–\(t\) graphs if \(v = 0\), \(s = 0\) when \(t = 0\). figure — add your own
  4. P12‑2dDetermine \(s\) and \(a\) when \(t = 3\) s if \(s = 0\) when \(t = 0\). figure — add your own
  5. P12‑2eDraw the \(v\)–\(t\) graph if \(v = 0\) when \(t = 0\). Find the equation \(v = f(t)\) for each segment. figure — add your own
  6. P12‑2fDetermine \(v\) at \(s = 2\) m if \(v = 1\) m/s at \(s = 0\). figure — add your own
  7. P12‑2gDetermine \(a\) at \(s = 1\) m. figure — add your own
Fundamental
  1. F12‑9The particle travels along a straight track such that its position is described by the \(s\)–\(t\) graph. Construct the \(v\)–\(t\) graph for the same time interval. figure — add your own
  2. F12‑10The sports car travels along a straight road such that its position is described by the graph. Construct the \(v\)–\(t\) and \(a\)–\(t\) graphs for \(0 \le t \le 10\) s. figure — add your own
  3. F12‑11A bicycle travels along a straight road where its velocity is described by the \(v\)–\(s\) graph. Construct the \(a\)–\(s\) graph for the same interval. figure — add your own
  4. F12‑12The sports car travels along a straight road such that its acceleration is described by the graph. Construct the \(v\)–\(s\) graph and specify the velocity when \(s = 10\) m and \(s = 15\) m. figure — add your own
  5. F12‑13The dragster starts from rest and has an acceleration described by the graph. Construct the \(v\)–\(t\) graph for \(0 \le t \le t'\), where \(t'\) is the time for the car to come to rest. figure — add your own
  6. F12‑14The dragster starts from rest and has a velocity described by the graph. Construct the \(s\)–\(t\) graph during \(0 \le t \le 15\) s. Also determine the total distance traveled. figure — add your own
  7. F12‑15A train starts from station \(A\) and for the first kilometer travels with uniform acceleration. Then for the next two kilometers it travels with uniform speed. Finally it decelerates uniformly for another kilometer before coming to rest at station \(B\). If the time for the whole journey is six minutes, draw the \(v\)–\(t\) graph and determine the maximum speed. figure — add your own
Problems
  1. 12‑35A train starts from station \(A\) and for the first kilometer it travels with uniform acceleration. Then for the next two kilometers it travels with uniform speed. Finally the train decelerates uniformly for another kilometer before coming to rest at station \(B\). If the time for the whole journey is six minutes, draw the \(v\)–\(t\) graph and determine the maximum speed. figure — add your own
  2. 12‑36If the position of a particle is \(s = [2\sin(\pi/5)t + 4]\) m, where \(t\) is in seconds, construct the \(s\)–\(t\), \(v\)–\(t\), and \(a\)–\(t\) graphs for \(0 \le t \le 10\) s.
  3. 12‑37A particle starts from \(s = 0\) and travels along a straight line with \(v = (t^2 - 4t + 3)\) m/s, where \(t\) is in seconds. Construct the \(v\)–\(t\) and \(a\)–\(t\) graphs for \(0 \le t \le 4\) s.
  4. 12‑38Two rockets start from rest at the same elevation. Rocket \(A\) accelerates vertically at \(20\) m/s² for \(12\) s and then maintains constant speed. Rocket \(B\) accelerates at \(15\) m/s² until reaching a constant speed of \(150\) m/s. Construct the \(a\)–\(t\), \(v\)–\(t\), and \(s\)–\(t\) graphs for each rocket until \(t = 20\) s. What is the distance between the rockets when \(t = 20\) s? figure — add your own
  5. 12‑39If the position of a particle is \(s = [3\sin(\pi/4)t + 8]\) m, where \(t\) is in seconds, construct the \(s\)–\(t\), \(v\)–\(t\), and \(a\)–\(t\) graphs for \(0 \le t \le 10\) s.
  6. 12‑40The \(s\)–\(t\) graph for a train has been experimentally determined. From the data construct the \(v\)–\(t\) and \(a\)–\(t\) graphs for \(0 \le t \le 40\) s. For \(0 \le t \le 30\) s, the curve is \(s = (0.4t^2)\) m, and then it becomes straight for \(t \ge 30\) s. figure — add your own
  7. 12‑41The velocity of a car is plotted as shown. Determine the total distance the car moves until it stops (\(t = 80\) s). Construct the \(a\)–\(t\) graph. figure — add your own
  8. 12‑42The snowmobile moves along a straight course according to the \(v\)–\(t\) graph. Construct the \(s\)–\(t\) and \(a\)–\(t\) graphs for the same 50‑s interval. When \(t = 0\), \(s = 0\). figure — add your own
  9. 12‑43The \(v\)–\(t\) graph for a particle moving through an electric field from one plate to another has the shape shown. The acceleration and deceleration are constant and both have magnitude \(4\) m/s². If the plates are spaced \(200\) mm apart, determine the maximum velocity \(v_{\max}\) and the time \(t'\) for the particle to travel from one plate to the other. Also draw the \(s\)–\(t\) graph. When \(t = t'/2\) the particle is at \(s = 100\) mm. figure — add your own
  10. 12‑44The \(v\)–\(t\) graph for a particle moving through an electric field has \(t' = 0.2\) s and \(v_{\max} = 10\) m/s. Draw the \(s\)–\(t\) and \(a\)–\(t\) graphs. When \(t = t'/2\) the particle is at \(s = 0.5\) m. figure — add your own
  11. 12‑45The motion of a jet plane just after landing is described by the \(a\)–\(t\) graph. Determine the time \(t'\) when the jet stops. Construct the \(v\)–\(t\) and \(s\)–\(t\) graphs. Here \(s = 0\), and \(v = 150\) m/s when \(t = 0\). figure — add your own
  12. 12‑46A two‑stage rocket is fired vertically from rest at \(s = 0\) with the acceleration as shown. After \(30\) s the first stage burns out and the second stage ignites. Plot the \(v\)–\(t\) and \(s\)–\(t\) graphs for the second stage for \(0 \le t \le 60\) s. figure — add your own
  13. 12‑47The \(a\)–\(t\) graph of the bullet train is shown. If the train starts from rest, determine the elapsed time \(t'\) before it again comes to rest. What is the total distance traveled? Construct the \(v\)–\(t\) and \(s\)–\(t\) graphs. figure — add your own
  14. 12‑48The \(v\)–\(t\) graph for a train has been experimentally determined. From the data construct the \(s\)–\(t\) and \(a\)–\(t\) graphs for \(0 \le t \le 180\) s. When \(t = 0\), \(s = 0\). figure — add your own
  15. 12‑49The \(v\)–\(s\) graph for a go‑cart traveling on a straight road is shown. Determine the acceleration at \(s = 50\) m and \(s = 150\) m. Draw the \(a\)–\(s\) graph. figure — add your own
  16. 12‑50The jet car is originally traveling at \(10\) m/s when it is subjected to the acceleration shown. Determine the car's maximum velocity and the time \(t'\) when it stops. When \(t = 0\), \(s = 0\). figure — add your own
  17. 12‑51The race car starts from rest and travels along a straight road until it reaches \(26\) m/s in \(8\) s as shown on the \(v\)–\(t\) graph. The flat part is caused by shifting gears. Draw the \(a\)–\(t\) graph and determine the maximum acceleration. figure — add your own
  18. 12‑52A car starts from rest and travels along a straight road with a velocity described by the graph. Determine the total distance traveled until the car stops. Construct the \(s\)–\(t\) and \(a\)–\(t\) graphs. figure — add your own
  19. 12‑53The \(v\)–\(s\) graph for an airplane traveling on a straight runway is shown. Determine the acceleration at \(s = 100\) m and \(s = 150\) m. Draw the \(a\)–\(s\) graph. figure — add your own
  20. 12‑54The \(a\)–\(s\) graph for a jeep traveling along a straight road is given for the first \(300\) m. Construct the \(v\)–\(s\) graph. At \(s = 0\), \(v = 0\). figure — add your own
  21. 12‑55The \(v\)–\(t\) graph for the motion of a car is shown. Draw the \(s\)–\(t\) and \(a\)–\(t\) graphs. Also determine the average speed and the distance traveled for the 15‑s interval. When \(t = 0\), \(s = 0\). figure — add your own
  22. 12‑56A motorcycle starts from rest at \(s = 0\) and travels along a straight road with the speed shown by the \(v\)–\(t\) graph. Determine the total distance traveled until it stops when \(t = 15\) s. Also plot the \(a\)–\(t\) and \(s\)–\(t\) graphs. figure — add your own
  23. 12‑57A motorcycle starts from rest at \(s = 0\) and travels along a straight road with the speed shown by the \(v\)–\(t\) graph. Determine the motorcycle's acceleration and position when \(t = 8\) s and \(t = 12\) s. figure — add your own
12.3 Curvilinear Motion ~100 problems
Preliminary
  1. P12‑3aUse the chain‑rule and find \(\dot{y}\) and \(\ddot{y}\) in terms of \(x, \dot{x}, \ddot{x}\) if \(y = 4x^2\).
  2. P12‑3bUse the chain‑rule and find \(\dot{y}\) and \(\ddot{y}\) in terms of \(x, \dot{x}, \ddot{x}\) if \(y = 3e^x\).
  3. P12‑3cUse the chain‑rule and find \(\dot{y}\) and \(\ddot{y}\) in terms of \(x, \dot{x}, \ddot{x}\) if \(y = 6\sin x\).
  4. P12‑4The particle travels from \(A\) to \(B\). Identify the three unknowns and write the three equations needed to solve for them. figure — add your own
  5. P12‑5The particle travels from \(A\) to \(B\). Identify the three unknowns and write the three equations needed to solve for them. figure — add your own
  6. P12‑6The particle travels from \(A\) to \(B\). Identify the three unknowns and write the three equations needed to solve for them. figure — add your own
  7. P12‑7aDetermine the acceleration at the instant shown. figure — add your own
  8. P12‑7bDetermine the increase in speed and the normal component of acceleration at \(s = 2\) m. At \(s = 0\), \(v = 0\). figure — add your own
  9. P12‑7cDetermine the acceleration at the instant shown. The particle has a constant speed of \(2\) m/s. figure — add your own
  10. P12‑7dDetermine the normal and tangential components of acceleration at \(s = 0\) if \(v = (4s + 1)\) m/s, where \(s\) is in meters. figure — add your own
  11. P12‑7eDetermine the acceleration at \(s = 2\) m if \(\dot{v} = (2s)\) m/s², where \(s\) is in meters. At \(s = 0\), \(v = 1\) m/s. figure — add your own
  12. P12‑7fDetermine the acceleration when \(t = 1\) s if \(v = (4t^2 + 2)\) m/s, where \(t\) is in seconds. figure — add your own
Fundamental
  1. F12‑16A particle is traveling along the straight path. If its position along the \(x\) axis is \(x = (8t)\) m, where \(t\) is in seconds, determine its speed when \(t = 2\) s. figure — add your own
  2. F12‑17If the \(x\) and \(y\) components of a particle's velocity are \(v_x = (32t)\) m/s and \(v_y = 8\) m/s, determine the equation of the path \(y = f(x)\), if \(x = 0\) and \(y = 0\) when \(t = 0\). figure — add your own
  3. F12‑18A particle travels along a straight‑line path \(y = 0.5x\). If \(v_x = (2t^2)\) m/s, where \(t\) is in seconds, determine the magnitude of the velocity and acceleration when \(t = 4\) s. figure — add your own
  4. F12‑19A particle is traveling along the parabolic path \(y = 0.25x^2\). If \(x = 8\) m, \(v_x = 8\) m/s, and \(a_x = 4\) m/s² when \(t = 2\) s, determine the magnitude of the velocity and acceleration at this instant. figure — add your own
  5. F12‑20The box slides down the slope described by \(y = (0.05x^2)\) m, where \(x\) is in meters. If the box has \(v_x = -3\) m/s and \(a_x = -1.5\) m/s² at \(x = 5\) m, determine the \(y\) components of velocity and acceleration. figure — add your own
  6. F12‑21The ball is kicked from point \(A\) with \(v_A = 10\) m/s. Determine the maximum height \(h\) it reaches. figure — add your own
  7. F12‑22The ball is kicked from point \(A\) with \(v_A = 10\) m/s. Determine the range \(R\) and the speed when the ball strikes the ground. figure — add your own
  8. F12‑23Determine the speed at which the basketball at \(A\) must be thrown at \(30^\circ\) so that it makes it to the basket at \(B\). figure — add your own
  9. F12‑24Water is sprayed at \(90^\circ\) from the slope at \(20\) m/s. Determine the range \(R\). figure — add your own
  10. F12‑25A ball is thrown from \(A\). If it is required to clear the wall at \(B\), determine the minimum magnitude of its initial velocity \(v_A\). figure — add your own
  11. F12‑26A projectile is fired with \(v_A = 150\) m/s off the roof of the building. Determine the range \(R\) where it strikes the ground at \(B\). figure — add your own
  12. F12‑27The boat is traveling along the circular path with \(v = (0.0625t^2)\) m/s, where \(t\) is in seconds. Determine the magnitude of its acceleration when \(t = 10\) s. figure — add your own
  13. F12‑28The car is traveling along the road with \(v = (2s)\) m/s, where \(s\) is in meters. Determine the magnitude of its acceleration when \(s = 10\) m. figure — add your own
  14. F12‑29If the car decelerates uniformly along the curved road from \(25\) m/s at \(A\) to \(15\) m/s at \(C\), determine the acceleration at \(B\). figure — add your own
  15. F12‑30When \(x = 3\) m, the crate has a speed of \(6\) m/s which is increasing at \(2\) m/s². Determine the direction of the crate's velocity and the magnitude of its acceleration at this instant. figure — add your own
  16. F12‑31If the motorcycle has a deceleration of \(a_t = -(0.001s)\) m/s² and its speed at \(A\) is \(25\) m/s, determine the magnitude of its acceleration when it passes point \(B\). figure — add your own
  17. F12‑32The car travels up the hill with \(v = (0.2s)\) m/s, where \(s\) is measured from \(A\). Determine the magnitude of its acceleration at \(s = 50\) m, where \(\rho = 500\) m. figure — add your own
  18. F12‑33The car has a speed of \(15\) m/s. Determine the angular velocity \(\dot{\theta}\) of the radial line \(OA\) at this instant. figure — add your own
  19. F12‑34The platform is rotating about the vertical axis such that \(\theta = (4t^{3/2})\) rad, where \(t\) is in seconds. A ball rolls outward along the radial groove so that \(r = (0.1t^3)\) m, where \(t\) is in seconds. Determine the magnitudes of the velocity and acceleration when \(t = 1.5\) s. figure — add your own
  20. F12‑35Peg \(P\) is driven by the fork link \(OA\) along the curved path \(r = (2\theta)\) m. At \(\theta = \pi/4\) rad, \(\dot{\theta} = 3\) rad/s and \(\ddot{\theta} = 1\) rad/s². Determine the magnitude of the peg's acceleration. figure — add your own
  21. F12‑36Peg \(P\) is driven by the forked link \(OA\) along \(r = e^\theta\), where \(r\) is in meters. When \(\theta = \pi/4\) rad, \(\dot{\theta} = 2\) rad/s and \(\ddot{\theta} = 4\) rad/s². Determine the radial and transverse components of the peg's acceleration. figure — add your own
  22. F12‑37The collars are pin connected at \(B\) and are free to move along rod \(OA\) and the curved guide \(r = [0.2(1 + \cos\theta)]\) m. At \(\theta = 30^\circ\), \(\dot{\theta} = 3\) rad/s. Determine the magnitude of the velocity of the collars. figure — add your own
  23. F12‑38At \(\theta = 45^\circ\), the athlete is running with a constant speed of \(2\) m/s. Determine the angular velocity at which the camera must turn to follow the motion. figure — add your own
Problems
  1. 12‑69The velocity of a particle is \(\mathbf{v} = \{16t^2\mathbf{i} + 4t^3\mathbf{j} + (5t + 2)\mathbf{k}\}\) m/s, where \(t\) is in seconds. If the particle is at the origin when \(t = 0\), determine the magnitude of the acceleration when \(t = 2\) s. Also, what are the \(x, y, z\) coordinates at this instant?
  2. 12‑70The position of a particle is \(\mathbf{r} = [5(\cos 2t)\mathbf{i} + 4(\sin 2t)\mathbf{j}]\) m, where \(t\) is in seconds and the arguments are in radians. Determine the magnitudes of velocity and acceleration when \(t = 1\) s. Also prove that the path is elliptical.
  3. 12‑71The velocity of a particle is \(\mathbf{v} = \{3\mathbf{i} + (6 - 2t)\mathbf{j}\}\) m/s, where \(t\) is in seconds. If \(\mathbf{r} = \mathbf{0}\) when \(t = 0\), determine the displacement from \(t = 1\) s to \(t = 3\) s.
  4. 12‑72If the velocity of a particle is \(\mathbf{v}(t) = [0.8t^2\mathbf{i} + 12t^{1/2}\mathbf{j} + 5\mathbf{k}]\) m/s, determine the magnitude and coordinate direction angles of the acceleration when \(t = 2\) s.
  5. 12‑73When a rocket reaches an altitude of \(40\) m it begins to travel along the parabolic path \((y - 40)^2 = 160x\), where coordinates are in meters. If \(v_y\) is constant at \(180\) m/s, determine the magnitudes of velocity and acceleration when it reaches an altitude of \(80\) m. figure — add your own
  6. 12‑74A particle is traveling with \(\mathbf{v} = \{3\sqrt{t}e^{-0.2t}\mathbf{i} + 4e^{-0.8t}\mathbf{j}\}\) m/s, where \(t\) is in seconds. Determine the magnitude of the displacement from \(t = 0\) to \(t = 3\) s using Simpson's rule with \(n = 100\). What is the magnitude of acceleration when \(t = 2\) s?
  7. 12‑75A particle travels along the curve from \(A\) to \(B\) in \(2\) s. It takes \(4\) s from \(B\) to \(C\) and \(3\) s from \(C\) to \(D\). Determine its average speed from \(A\) to \(D\). figure — add your own
  8. 12‑76A particle travels along the curve from \(A\) to \(B\) in \(5\) s, \(8\) s from \(B\) to \(C\), and \(10\) s from \(C\) to \(A\). Determine its average speed around the closed path. figure — add your own
  9. 12‑77The position of a crate sliding down a ramp is \(x = (0.25t^3)\) m, \(y = (1.5t^2)\) m, \(z = (6 - 0.75t^{5/2})\) m, where \(t\) is in seconds. Determine the magnitude of the crate's velocity and acceleration when \(t = 2\) s.
  10. 12‑78A rocket is fired from rest at \(x = 0\) and travels along \(y^2 = [120(10^3)x]\) m. If \(a_x = (\frac{1}{4}t^2)\) m/s², where \(t\) is in seconds, determine the magnitude of the velocity and acceleration when \(t = 10\) s.
  11. 12‑79The particle travels along \(y = 0.5x^2\). If \(v_x = (5t)\) m/s, where \(t\) is in seconds, determine the particle's distance from the origin and the magnitude of its acceleration when \(t = 1\) s. When \(t = 0\), \(x = 0\), \(y = 0\). figure — add your own
  12. 12‑80The motorcycle travels with constant speed \(v_0\) along a path that for a short distance takes the form of a sine curve. Determine the \(x\) and \(y\) components of its velocity at any instant. figure — add your own
  13. 12‑81A particle travels along the curve from \(A\) to \(B\) in \(1\) s. If it takes \(3\) s from \(A\) to \(C\), determine its average velocity from \(B\) to \(C\). figure — add your own
  14. 12‑82The roller coaster car travels down the helical path at constant speed such that \(x = c\sin kt\), \(y = c\cos kt\), \(z = h - bt\), where \(c, h, b\) are constants. Determine the magnitudes of its velocity and acceleration. figure — add your own
  15. 12‑83The flight path of the helicopter as it takes off from \(A\) is \(x = (2t^2)\) m and \(y = (0.04t^3)\) m, where \(t\) is in seconds. Determine the distance from point \(A\) and the magnitudes of velocity and acceleration when \(t = 10\) s. figure — add your own
  16. 12‑84Pegs \(A\) and \(B\) are restricted to move in elliptical slots due to the motion of the slotted link. If the link moves with a constant speed of \(10\) m/s, determine the magnitude of the velocity and acceleration of peg \(A\) when \(x = 1\) m. figure — add your own
  17. 12‑85It is observed that the time for the ball to strike the ground at \(B\) is \(2.5\) s. Determine the speed \(v_A\) and angle \(\theta_A\) at which the ball was thrown. figure — add your own
  18. 12‑86Neglecting the size of the ball, determine the magnitude \(v_A\) of the basketball's initial velocity and its velocity when it passes through the basket. figure — add your own
  19. 12‑87A projectile is fired from the platform at \(B\). The shooter fires from point \(A\) at an angle of \(30^\circ\). Determine the muzzle speed of the bullet if it hits the projectile at \(C\). figure — add your own
  20. 12‑88Determine the minimum initial velocity \(v_0\) and the corresponding angle \(\theta_0\) at which the ball must be kicked in order to just cross over the \(3\)-m-high fence. figure — add your own
  21. 12‑89A projectile is given a velocity \(\mathbf{v}_0\) at an angle \(\phi\) above the horizontal. Determine the distance \(d\) to where it strikes the sloped ground. The acceleration due to gravity is \(g\). figure — add your own
  22. 12‑90A projectile is given a velocity \(\mathbf{v}_0\). Determine the angle \(\phi\) at which it should be launched so that \(d\) is a maximum. The acceleration due to gravity is \(g\). figure — add your own
  23. 12‑91The girl at \(A\) can throw a ball at \(v_A = 10\) m/s. Calculate the maximum possible range \(R = R_{\max}\) and the associated angle \(\theta\) at which it should be thrown. Assume the ball is caught at \(B\) at the same elevation. figure — add your own
  24. 12‑92Show that the girl at \(A\) can throw the ball to the boy at \(B\) by launching it at equal angles measured up or down from a \(45^\circ\) inclination. If \(v_A = 10\) m/s, determine the range \(R\) if this value is \(15^\circ\), i.e., \(\theta_1 = 30^\circ\) and \(\theta_2 = 60^\circ\). figure — add your own
  25. 12‑93The boy at \(A\) attempts to throw a ball over the roof of a barn with \(v_A = 15\) m/s. Determine the angle \(\theta_A\) at which the ball must be thrown so that it reaches its maximum height at \(C\). Also find the distance \(d\) where the boy should stand. figure — add your own
  26. 12‑94The boy at \(A\) attempts to throw a ball over the roof of a barn such that it is launched at \(\theta_A = 40^\circ\). Determine the minimum speed \(v_A\) at which he must throw the ball so that it reaches its maximum height at \(C\). Also find the distance \(d\). figure — add your own
  27. 12‑95Measurements of a shot recorded on videotape during a basketball game are shown. The ball passed through the hoop even though it barely cleared the hands of player \(B\). Neglecting the size of the ball, determine the magnitude \(v_A\) of its initial velocity and the height \(h\) when it passes over player \(B\). figure — add your own
  28. 12‑96The golf ball is hit at \(A\) with \(v_A = 40\) m/s at \(30^\circ\) with the horizontal as shown. Determine the distance \(d\) where the ball strikes the slope at \(B\). figure — add your own
  29. 12‑97It is observed that the skier leaves the ramp \(A\) at \(\theta_A = 25^\circ\) with the horizontal. If he strikes the ground at \(B\), determine his initial speed \(v_A\) and the time of flight \(t_{AB}\). figure — add your own
  30. 12‑98It is observed that the skier leaves the ramp \(A\) at \(\theta_A = 25^\circ\) with the horizontal. If he strikes the ground at \(B\), determine his initial speed \(v_A\) and the speed at which he strikes the ground. figure — add your own
  31. 12‑99The projectile is launched with velocity \(\mathbf{v}_0\). Determine the range \(R\), the maximum height \(h\), and the time of flight. Express the results in terms of \(\theta\) and \(v_0\). The acceleration due to gravity is \(g\). figure — add your own
  32. 12‑100The missile at \(A\) takes off from rest and rises vertically to \(B\), where its fuel runs out in \(8\) s. If the acceleration varies with time as shown, determine the missile's height \(h_B\) and speed \(v_B\). If by internal controls the missile is then suddenly pointed \(45^\circ\) and allowed to travel in free flight, determine the maximum height \(h_C\) and the range \(R\) to where it crashes at \(D\). figure — add your own
  33. 12‑101The velocity of the water jet discharging from the orifice is \(v = \sqrt{2gh}\), where \(h = 2\) m is the depth of the orifice from the free water surface. Determine the time for a particle of water leaving the orifice to reach point \(B\) and the horizontal distance \(x\) where it hits the surface. figure — add your own
  34. 12‑102The man at \(A\) wishes to throw two darts at the target at \(B\) so that they arrive at the same time. If each dart is thrown with \(10\) m/s, determine the angles \(\theta_C\) and \(\theta_D\) at which they should be thrown and the time between each throw. The first dart must be thrown at \(\theta_C > \theta_D\), then the second at \(\theta_D\). figure — add your own
  35. 12‑103If the dart is thrown with \(10\) m/s, determine the shortest possible time before it strikes the target. Also, what is the corresponding angle \(\theta_A\) and the velocity of the dart when it strikes the target? figure — add your own
  36. 12‑104If the dart is thrown with \(10\) m/s, determine the longest possible time when it strikes the target. Also, what is the corresponding angle \(\theta_A\) and the velocity of the dart when it strikes the target? figure — add your own
  37. 12‑105The drinking fountain is designed such that the nozzle is located from the edge of the basin as shown. Determine the maximum and minimum speed at which water can be ejected from the nozzle so that it does not splash over the sides at \(B\) and \(C\). figure — add your own
  38. 12‑106The balloon \(A\) is ascending at \(v_A = 12\) km/h and is being carried horizontally by the wind at \(v_w = 20\) km/h. If a ballast bag is dropped from the balloon at \(h = 50\) m, determine the time needed for it to strike the ground. Assume the bag was released with the same velocity as the balloon. Also, with what speed does the bag strike the ground? figure — add your own
  39. 12‑107The snowmobile is traveling at \(10\) m/s when it leaves the embankment at \(A\). Determine the time of flight from \(A\) to \(B\) and the range \(R\). figure — add your own
  40. 12‑108A boy throws a ball at \(O\) in the air with speed \(v_0\) at angle \(\theta_1\). If he then throws another ball with the same speed \(v_0\) at angle \(\theta_2 < \theta_1\), determine the time between throws so that the balls collide in midair at \(B\). figure — add your own
  41. 12‑109Small packages traveling on the conveyor belt fall off into a \(1\)-m-long loading car. If the conveyor is running at \(v_C = 2\) m/s, determine the smallest and largest distance \(R\) at which the end \(A\) of the car may be placed so that the packages enter the car. figure — add your own
  42. 12‑110The motion of a particle is \(x = (2t + t^2)\) m and \(y = (t^2)\) m, where \(t\) is in seconds. Determine the normal and tangential components of velocity and acceleration when \(t = 2\) s.
  43. 12‑111Determine the maximum constant speed a race car can have if the acceleration cannot exceed \(7.5\) m/s² while rounding a track with radius of curvature \(200\) m.
  44. 12‑112A particle moves along \(y = \sin x\) with constant speed \(v = 2\) m/s. Determine the normal and tangential components of its velocity and acceleration at any instant.
  45. 12‑113The position of a particle is \(\mathbf{r} = [4(t - \sin t)\mathbf{i} + (2t^2 - 3)\mathbf{j}]\) m, where \(t\) is in seconds and the argument for sine is in radians. Determine the speed and the normal and tangential components of acceleration when \(t = 1\) s.
  46. 12‑114The car travels along a curve with radius \(300\) m. If its speed is uniformly increased from \(15\) m/s to \(27\) m/s in \(3\) s, determine the magnitude of its acceleration when the speed is \(20\) m/s. figure — add your own
  47. 12‑115When the car reaches point \(A\) it has speed \(25\) m/s. If the brakes are applied, its speed is reduced by \(a_t = -\frac{1}{4}t^{1/2}\) m/s². Determine the magnitude of acceleration just before it reaches point \(C\). figure — add your own
  48. 12‑116When the car reaches point \(A\), it has speed \(25\) m/s. If the brakes are applied, its speed is reduced by \(a_t = (0.001s - 1)\) m/s². Determine the magnitude of acceleration just before it reaches point \(C\). figure — add your own
  49. 12‑117At a given instant, a car travels along a circular curved road with speed \(20\) m/s while decreasing its speed at \(3\) m/s². If the magnitude of the car's acceleration is \(5\) m/s², determine the radius of curvature.
  50. 12‑118Car \(B\) turns such that its speed is increased by \((a_t)_B = (0.5e^t)\) m/s², where \(t\) is in seconds. If the car starts from rest when \(\theta = 0^\circ\), determine the magnitudes of its velocity and acceleration when the arm \(AB\) rotates \(\theta = 30^\circ\). Neglect the size of the car. figure — add your own
  51. 12‑119The motorcycle is traveling at \(1\) m/s when at \(A\). If the speed is then increased at \(\dot{v} = 0.1\) m/s², determine its speed and acceleration at \(t = 5\) s. figure — add your own
  52. 12‑120The car passes point \(A\) with speed \(25\) m/s after which its speed is \(v = (25 - 0.15s)\) m/s. Determine the magnitude of acceleration when it reaches point \(B\), where \(s = 51.5\) m and \(x = 50\) m. figure — add your own
  53. 12‑121If the car passes point \(A\) with speed \(20\) m/s and begins to increase its speed at \(a_t = 0.5\) m/s², determine the magnitude of acceleration when \(s = 101.68\) m and \(x = 0\). figure — add your own
  54. 12‑122The car travels along the circular path such that its speed is increased by \(a_t = (0.5e^t)\) m/s², where \(t\) is in seconds. Determine the magnitudes of velocity and acceleration after the car has traveled \(s = 18\) m starting from rest. Neglect the size of the car. figure — add your own
  55. 12‑123The satellite \(S\) travels around the earth in a circular path with constant speed \(20\) Mm/h. If the acceleration is \(2.5\) m/s², determine the altitude \(h\). Assume the earth's diameter is \(12.713\) km. figure — add your own
  56. 12‑124The car has initial speed \(v_0 = 20\) m/s. If it increases its speed along the circular track at \(s = 0\), \(a_t = (0.8s)\) m/s², where \(s\) is in meters, determine the time needed for the car to travel \(s = 25\) m. figure — add your own
  57. 12‑125The car starts from rest at \(s = 0\) and increases its speed at \(a_t = 4\) m/s². Determine the time when the magnitude of acceleration becomes \(20\) m/s². At what position \(s\) does this occur? figure — add your own
  58. 12‑126At a given instant the train engine at \(E\) has speed \(20\) m/s and acceleration \(14\) m/s² acting in the direction shown. Determine the rate of increase in the train's speed and the radius of curvature \(\rho\). figure — add your own
  59. 12‑127When the roller coaster is at \(B\), it has speed \(25\) m/s, increasing at \(a_t = 3\) m/s². Determine the magnitude of acceleration at this instant and the direction angle it makes with the \(x\) axis. figure — add your own
  60. 12‑128If the roller coaster starts from rest at \(A\) and its speed increases at \(a_t = (6 - 0.06s)\) m/s², determine the magnitude of its acceleration when it reaches \(B\) where \(s_B = 40\) m. figure — add your own
  61. 12‑129The box of negligible size is sliding down along a curved path defined by \(y = 0.4x^2\). When at \(A(x_A = 2\) m, \(y_A = 1.6\) m), the speed is \(8\) m/s and \(dv/dt = 4\) m/s². Determine the magnitude of the acceleration. figure — add your own
  62. 12‑130The position of a particle traveling along a curved path is \(s = (3t^3 - 4t^2 + 4)\) m, where \(t\) is in seconds. When \(t = 2\) s, the particle is at a position where the radius of curvature is \(25\) m. Determine the magnitude of the particle's acceleration at this instant.
  63. 12‑131A particle travels around a circular path with radius \(50\) m. If it is initially traveling with speed \(10\) m/s and its speed increases at \(\dot{v} = (0.05v)\) m/s², determine the magnitude of the acceleration four seconds later.
  64. 12‑132The motorcycle is traveling at \(40\) m/s when at \(A\). If the speed is then decreased at \(\dot{v} = -(0.05s)\) m/s², where \(s\) is measured from \(A\), determine its speed and acceleration when it reaches \(B\). figure — add your own
  65. 12‑133The ball is ejected horizontally from the tube with speed \(8\) m/s. Find the equation of the path \(y = f(x)\) and then find the ball's velocity and the normal and tangential components of acceleration when \(t = 0.25\) s. figure — add your own
  66. 12‑134A boat is traveling along a circular path with radius \(20\) m. Determine the magnitude of the boat's acceleration when the speed is \(5\) m/s and the rate of increase in speed is \(\dot{v} = 2\) m/s².
  67. 12‑135Starting from rest, a bicyclist travels around a horizontal circular path \(\rho = 10\) m at \(v = (0.09t^2 + 0.1t)\) m/s, where \(t\) is in seconds. Determine the magnitudes of his velocity and acceleration when he has traveled \(s = 3\) m.
  68. 12‑136The motorcycle is traveling at a constant speed of \(60\) km/h. Determine the magnitude of its acceleration when it is at point \(A\). figure — add your own
  69. 12‑137When \(t = 0\), the train has speed \(8\) m/s, increasing at \(0.5\) m/s². Determine the magnitude of the acceleration of the engine when it reaches point \(A\), at \(t = 20\) s. Here \(\rho_A = 400\) m. figure — add your own
  70. 12‑138The ball is ejected horizontally from the tube with speed \(8\) m/s. Find the equation of the path \(y = f(x)\) and then find the ball's velocity and the normal and tangential components of acceleration when \(t = 0.25\) s. figure — add your own
  71. 12‑139The motorcycle travels along the elliptical track at constant speed \(v\). Determine its greatest acceleration if \(a > b\). figure — add your own
  72. 12‑140The motorcycle travels along the elliptical track at constant speed \(v\). Determine its smallest acceleration if \(a > b\). figure — add your own
  73. 12‑141The race car has initial speed \(v_A = 15\) m/s at \(A\). If it increases its speed along the circular track at \(a_t = (0.4s)\) m/s², where \(s\) is in meters, determine the time needed for the car to travel \(20\) m. Take \(\rho = 150\) m. figure — add your own
  74. 12‑142The ball is kicked with \(v_A = 8\) m/s at \(\theta_A = 40^\circ\) with the horizontal. Find the equation of the path \(y = f(x)\), and then determine the ball's velocity and the normal and tangential components of its acceleration when \(t = 0.25\) s. figure — add your own
  75. 12‑143Cars move around the "traffic circle" which is in the shape of an ellipse. If the speed limit is \(60\) km/h, determine the minimum acceleration experienced by the passengers. figure — add your own
  76. 12‑144Cars move around the "traffic circle" which is in the shape of an ellipse. If the speed limit is \(60\) km/h, determine the maximum acceleration experienced by the passengers. figure — add your own
  77. 12‑145The particle travels with a constant speed of \(300\) mm/s along the curve. Determine the particle's acceleration when it is located at point \((200\) mm, \(100\) mm) and sketch this vector on the curve. figure — add your own
  78. 12‑146The train passes point \(B\) with speed \(20\) m/s which is decreasing at \(a_t = -0.5\) m/s². Determine the magnitude of acceleration at this point. figure — add your own
  79. 12‑147The train passes point \(A\) with speed \(30\) m/s and begins to decrease its speed at \(a_t = -0.25\) m/s². Determine the magnitude of acceleration when it reaches point \(B\), where \(s_{AB} = 412\) m. figure — add your own
  80. 12‑148The jet plane is traveling with speed \(120\) m/s which is decreasing at \(40\) m/s² when it reaches point \(A\). Determine the magnitude of its acceleration at this point. Also specify the direction of flight measured from the \(x\) axis. figure — add your own
  81. 12‑149The jet plane is traveling with constant speed \(110\) m/s along the curved path. Determine the magnitude of the acceleration at the instant it reaches point \(A(y = 0)\). figure — add your own
  82. 12‑150Particles \(A\) and \(B\) are traveling counterclockwise around a circular track at constant speed \(8\) m/s. If at the instant shown the speed of \(A\) begins to increase by \((a_t)_A = (0.4s_A)\) m/s², where \(s_A\) is in meters, determine the distance measured counterclockwise along the track from \(B\) to \(A\) when \(t = 1\) s. What is the magnitude of the acceleration of each particle at this instant? figure — add your own
  83. 12‑151Particles \(A\) and \(B\) are traveling around a circular track at speed \(8\) m/s at the instant shown. If the speed of \(B\) is increasing by \((a_t)_B = 4\) m/s², and at the same instant \(A\) has an increase in speed of \((a_t)_A = 0.8t\) m/s², determine how long it takes for a collision to occur. What is the magnitude of the acceleration of each particle just before the collision? figure — add your own
  84. 12‑155If a particle's position is described by polar coordinates \(r = 4(1 + \sin t)\) m and \(\theta = (2e^{-t})\) rad, where \(t\) is in seconds and the argument for sine is in radians, determine the radial and transverse components of velocity and acceleration when \(t = 2\) s.
  85. 12‑156A particle travels around a limaçon \(r = b - a\cos\theta\), where \(a\) and \(b\) are constants. Determine the radial and transverse components of velocity and acceleration as a function of \(\theta\) and its time derivatives.
  86. 12‑157A particle moves along a circular path of radius \(300\) mm. If its angular velocity is \(\dot{\theta} = (2t^2)\) rad/s, where \(t\) is in seconds, determine the magnitude of the particle's acceleration when \(t = 2\) s.
  87. 12‑158For a short time a rocket travels up and to the right at constant speed \(800\) m/s along the parabolic path \(y = 600 - 35x^2\). Determine the radial and transverse components of velocity at the instant \(\theta = 60^\circ\), where \(\theta\) is measured counterclockwise from the \(x\) axis.
  88. 12‑159The box slides down the helical ramp with constant speed \(v = 2\) m/s. Determine the magnitude of its acceleration. The ramp descends \(1\) m for every full revolution. The mean radius is \(r = 0.5\) m. figure — add your own
  89. 12‑160The box slides down the helical ramp such that \(r = 0.5\) m, \(\theta = (0.5t^3)\) rad, and \(z = (2 - 0.2t^2)\) m, where \(t\) is in seconds. Determine the magnitudes of velocity and acceleration at the instant \(\theta = 2\pi\) rad. figure — add your own
  90. 12‑161If a particle's position is described by polar coordinates \(r = (2\sin 2\theta)\) m and \(\theta = (4t)\) rad, where \(t\) is in seconds, determine the radial and transverse components of velocity and acceleration when \(t = 1\) s.
  91. 12‑162If a particle moves along a path such that \(r = (e^{at})\) m and \(\theta = t\), where \(t\) is in seconds, plot the path \(r = f(\theta)\) and determine the radial and transverse components of velocity and acceleration.
  92. 12‑163A radar gun at \(O\) rotates with \(\dot{\theta} = 0.1\) rad/s and \(\ddot{\theta} = 0.025\) rad/s² at \(\theta = 45^\circ\) as it follows the motion of the car traveling along the circular road \(r = 200\) m. Determine the magnitudes of velocity and acceleration of the car. figure — add your own
  93. 12‑164The small washer is sliding down the cord \(OA\). When it is at the midpoint, its speed is \(28\) m/s and its acceleration is \(7\) m/s². Express the velocity and acceleration at this point in terms of cylindrical components. figure — add your own
  94. 12‑165The time rate of change of acceleration is referred to as the jerk. Calculate this vector in terms of cylindrical components using Eq. 12‑32.
  95. 12‑166A particle is moving along a circular path with radius \(400\) mm. Its position is \(\theta = (2t^2)\) rad, where \(t\) is in seconds. Determine the magnitude of the acceleration when \(\theta = 30^\circ\). The particle starts from rest when \(\theta = 0^\circ\).
  96. 12‑167The slotted link is pinned at \(O\) and as a result of the constant angular velocity \(\dot{\theta} = 3\) rad/s it drives peg \(P\) along the spiral guide \(r = (0.4\theta)\) m, where \(\theta\) is in radians. Determine the radial and transverse components of velocity and acceleration at \(\theta = \pi/3\) rad. figure — add your own
  97. 12‑168At the instant shown, the watersprinkler is rotating with \(\dot{\theta} = 2\) rad/s and \(\ddot{\theta} = 3\) rad/s². If the nozzle lies in the vertical plane and water is flowing through it at a constant rate of \(3\) m/s, determine the magnitudes of velocity and acceleration of a water particle as it exits the open end, \(r = 0.2\) m. figure — add your own
  98. 12‑169At the instant shown, the man is twirling a hose over his head with \(\dot{\theta} = 2\) rad/s and \(\ddot{\theta} = 3\) rad/s². If the hose lies in a horizontal plane and water flows at \(3\) m/s, determine the magnitudes of velocity and acceleration of a water particle as it exits the open end, \(r = 1.5\) m. figure — add your own
  99. 12‑170The partial surface of the cam is a logarithmic spiral \(r = (40e^{0.05\theta})\) mm, where \(\theta\) is in radians. If the cam is rotating at constant \(\dot{\theta} = 4\) rad/s, determine the magnitudes of velocity and acceleration of the follower rod at \(\theta = 30^\circ\). figure — add your own
  100. 12‑171Solve Prob. 12‑170 if the cam has \(\ddot{\theta} = 2\) rad/s² when \(\dot{\theta} = 4\) rad/s at \(\theta = 30^\circ\). figure — add your own
  101. 12‑172A cameraman standing at \(A\) is following a race car \(B\) traveling around a curved track at constant speed \(30\) m/s. Determine the angular rate \(\dot{\theta}\) at which the man must turn to keep the camera directed on the car at \(\theta = 30^\circ\). figure — add your own
  102. 12‑173The car travels around the circular track with constant speed \(20\) m/s. Determine the car's radial and transverse components of velocity and acceleration at \(\theta = \pi/4\) rad. figure — add your own
  103. 12‑174The car travels around the circular track such that \(\theta = (0.006t^2)\) rad, where \(t\) is in seconds. Determine the radial and transverse components of velocity and acceleration at \(t = 4\) s. figure — add your own
  104. 12‑175A block moves outward along the slot in the platform with \(\dot{r} = (4t)\) m/s, where \(t\) is in seconds. The platform rotates at constant \(6\) rad/s. If the block starts from rest at the center, determine the magnitudes of velocity and acceleration when \(t = 1\) s. figure — add your own
  105. 12‑176The pin follows the path \(r = (0.2 + 0.15\cos\theta)\) m. At \(\theta = 30^\circ\), \(\dot{\theta} = 0.7\) rad/s and \(\ddot{\theta} = 0.5\) rad/s². Determine the magnitudes of the pin's velocity and acceleration. Neglect the size of the pin. figure — add your own
  106. 12‑177The rod \(OA\) rotates clockwise with constant \(\dot{\theta} = 6\) rad/s. Two pin‑connected slider blocks at \(B\) move freely on \(OA\) and the curved rod \(r = 200(2 - \cos\theta)\) mm. Determine the speed of the slider blocks at \(\theta = 150^\circ\). figure — add your own
  107. 12‑178Determine the magnitude of the acceleration of the slider blocks in Prob. 12‑177 when \(\theta = 150^\circ\). figure — add your own
  108. 12‑179The rod \(OA\) rotates counterclockwise with constant \(\dot{\theta} = 5\) rad/s. Two pin‑connected slider blocks at \(B\) move freely on \(OA\) and the curved rod \(r = 100(2 - \cos\theta)\) mm. Determine the speed of the slider blocks at \(\theta = 120^\circ\). figure — add your own
  109. 12‑180Determine the magnitude of the acceleration of the slider blocks in Prob. 12‑179 when \(\theta = 120^\circ\). figure — add your own
  110. 12‑181The slotted arm \(AB\) drives pin \(C\) through the spiral groove \(r = a\theta\). If the angular velocity is constant at \(\dot{\theta}\), determine the radial and transverse components of velocity and acceleration.
  111. 12‑182The slotted arm \(AB\) drives pin \(C\) through \(r = (1.5\theta)\) m, where \(\theta\) is in radians. If the arm starts from rest when \(\theta = 60^\circ\) and is driven at \(\dot{\theta} = (4t)\) rad/s, where \(t\) is in seconds, determine the radial and transverse components of velocity and acceleration when \(t = 1\) s. figure — add your own
  112. 12‑183If the cam rotates clockwise with constant \(\dot{\theta} = 5\) rad/s, determine the magnitudes of velocity and acceleration of the follower rod \(AB\) at \(\theta = 30^\circ\). The surface is a limaçon \(r = (200 + 100\cos\theta)\) mm. figure — add your own
  113. 12‑184At \(\theta = 30^\circ\), the cam rotates with clockwise \(\dot{\theta} = 5\) rad/s and \(\ddot{\theta} = 6\) rad/s². Determine the magnitudes of velocity and acceleration of the follower rod \(AB\). The surface is \(r = (200 + 100\cos\theta)\) mm. figure — add your own
  114. 12‑185A truck is traveling along the horizontal circular curve of radius \(r = 60\) m with constant speed \(v = 20\) m/s. Determine the angular rate \(\dot{\theta}\) of the radial line \(r\) and the magnitude of the truck's acceleration. figure — add your own
  115. 12‑186A truck is traveling along the horizontal circular curve of radius \(r = 60\) m with speed \(20\) m/s which is increasing at \(3\) m/s². Determine the truck's radial and transverse components of acceleration. figure — add your own
  116. 12‑187The double collar \(C\) is pin connected such that one collar slides over the fixed rod and the other over the rotating rod \(AB\). If \(\dot{\theta} = (e^{0.5t^2})\) rad/s, where \(t\) is in seconds, and the fixed rod is \(r = [(0.4\sin\theta + 0.2)]\) m, determine the radial and transverse components of velocity and acceleration when \(t = 1\) s. When \(t = 0\), \(\theta = 0\). Use Simpson's rule with \(n = 50\) to determine \(\theta\) at \(t = 1\) s. figure — add your own
  117. 12‑188The double collar \(C\) is pin connected such that one collar slides over the fixed rod and the other over the rotating rod \(AB\). If the mechanism is to be designed so that the largest speed given to the collar is \(6\) m/s, determine the required constant angular velocity \(\dot{\theta}\) of rod \(AB\). The fixed rod is \(r = (0.4\sin\theta + 0.2)\) m. figure — add your own
  118. 12‑189For a short distance the train travels along a spiral \(r = (1000/\theta)\) m, where \(\theta\) is in radians. If it maintains constant speed \(v = 20\) m/s, determine the radial and transverse components of its velocity when \(\theta = (9\pi/4)\) rad. figure — add your own
  119. 12‑190For a short distance the train travels along a spiral \(r = (1000/\theta)\) m, where \(\theta\) is in radians. If the angular rate is constant \(\dot{\theta} = 0.2\) rad/s, determine the radial and transverse components of velocity and acceleration when \(\theta = (9\pi/4)\) rad. figure — add your own
  120. 12‑191The driver of the car maintains a constant speed of \(40\) m/s. Determine the angular velocity of the camera tracking the car when \(\theta = 15^\circ\). figure — add your own
  121. 12‑192When \(\theta = 15^\circ\), the car has speed \(50\) m/s which is increasing at \(6\) m/s². Determine the angular velocity of the camera tracking the car at this instant. figure — add your own
  122. 12‑193If the circular plate rotates clockwise with constant \(\dot{\theta} = 1.5\) rad/s, determine the magnitudes of velocity and acceleration of the follower rod \(AB\) when \(\theta = 2\pi/3\) rad. figure — add your own
  123. 12‑194When \(\theta = 2\pi/3\) rad, the angular velocity and angular acceleration of the circular plate are \(\dot{\theta} = 1.5\) rad/s and \(\ddot{\theta} = 3\) rad/s². Determine the magnitudes of velocity and acceleration of the rod \(AB\) at this instant. figure — add your own
12.4 Relative Motion ~30 problems
Fundamental
  1. F12‑39Determine the velocity of block \(D\) if end \(A\) of the rope is pulled down with \(v_A = 3\) m/s. figure — add your own
  2. F12‑40Determine the velocity of block \(A\) if end \(B\) of the rope is pulled down with \(6\) m/s. figure — add your own
  3. F12‑41Determine the velocity of block \(A\) if end \(B\) of the rope is pulled down with \(1.5\) m/s. figure — add your own
  4. F12‑42Determine the velocity of block \(A\) if end \(F\) of the rope is pulled down with \(v_F = 3\) m/s. figure — add your own
  5. F12‑43Determine the velocity of car \(A\) if point \(P\) on the cable has a speed of \(4\) m/s when the motor \(M\) winds the cable in. figure — add your own
  6. F12‑44Determine the velocity of block \(A\) if end \(B\) of the rope is pulled down with \(6\) m/s. figure — add your own
  7. F12‑45Car \(A\) is traveling with constant speed \(80\) km/h due north, while car \(B\) is traveling with constant speed \(100\) km/h due east. Determine the velocity of car \(B\) relative to car \(A\). figure — add your own
  8. F12‑46Two planes \(A\) and \(B\) are traveling with the constant velocities shown. Determine the magnitude and direction of the velocity of plane \(B\) relative to plane \(A\). figure — add your own
  9. F12‑47At the instant shown, cars \(A\) and \(B\) are traveling at the speeds shown. If \(B\) is accelerating at \(1200\) km/h² while \(A\) maintains constant speed, determine the velocity and acceleration of \(A\) with respect to \(B\). figure — add your own
  10. F12‑48At the instant shown, cars \(A\) and \(B\) are traveling at the speeds shown. If \(B\) is accelerating at \(1200\) km/h² while \(A\) maintains constant speed, determine the velocity and acceleration of \(A\) with respect to \(B\). figure — add your own
Problems
  1. 12‑195If the end of the cable at \(A\) is pulled down with \(5\) m/s, determine the speed at which block \(B\) rises. figure — add your own
  2. 12‑196Determine the displacement of the log if the truck at \(C\) pulls the cable \(1.2\) m to the right. figure — add your own
  3. 12‑197If end \(A\) of the cable is moving upward at \(v_A = 14\) m/s, determine the speed of block \(B\). figure — add your own
  4. 12‑198Determine the constant speed at which the cable at \(A\) must be drawn in by the motor in order to hoist the load \(6\) m in \(1.5\) s. figure — add your own
  5. 12‑199Starting from rest, the cable can be wound onto the drum at \(v_A = (3t^2)\) m/s, where \(t\) is in seconds. Determine the time needed to lift the load \(7\) m. figure — add your own
  6. 12‑200If the end of the cable at \(A\) is pulled down with \(2\) m/s, determine the speed at which block \(B\) rises. figure — add your own
  7. 12‑201The motor at \(C\) pulls in the cable with \(a_C = (3t^2)\) m/s², where \(t\) is in seconds. The motor at \(D\) draws in its cable at \(a_D = 5\) m/s². If both motors start from rest when \(d = 3\) m, determine (a) the time needed for \(d = 0\), and (b) the velocities of blocks \(A\) and \(B\) when this occurs. figure — add your own
  8. 12‑202Determine the speed of the block at \(B\). figure — add your own
  9. 12‑203If block \(A\) is moving downward with \(2\) m/s while \(C\) is moving up at \(1\) m/s, determine the speed of block \(B\). figure — add your own
  10. 12‑204Determine the speed of block \(A\) if the end of the rope is pulled down with \(4\) m/s. figure — add your own
  11. 12‑205If end \(A\) of the cable is moving at \(v_A = 3\) m/s, determine the speed of block \(B\). figure — add your own
  12. 12‑206The motor draws in the cable at \(C\) with constant velocity \(v_C = 4\) m/s. The motor draws in the cable at \(D\) with \(v_D = 3\) m/s. If \(d = 0\) when \(t = 0\), determine (a) the time needed for block \(A\) to rise \(3\) m, and (b) the relative velocity of block \(A\) with respect to block \(B\) when this occurs. figure — add your own
  13. 12‑207Determine the time needed for the load at \(B\) to attain a speed of \(10\) m/s, starting from rest, if the cable is drawn into the motor with an acceleration of \(3\) m/s². figure — add your own
  14. 12‑208The cable at \(A\) is being drawn toward the motor at \(v_A = 8\) m/s. Determine the velocity of the block. figure — add your own
  15. 12‑209If the hydraulic cylinder \(H\) draws in rod \(BC\) at \(1\) m/s, determine the speed of slider \(A\). figure — add your own
  16. 12‑210If the truck travels at constant speed \(v_T = 1.8\) m/s, determine the speed of the crate for any angle \(\theta\) of the rope. The rope has length \(30\) m and passes over a pulley at \(A\). Relate the coordinates and take the time derivative. figure — add your own
  17. 12‑211The crate \(C\) is being lifted by moving the roller at \(A\) downward with constant speed \(v_A = 2\) m/s along the guide. Determine the velocity and acceleration of the crate at the instant \(s = 1\) m. When the roller is at \(B\), the crate rests on the ground. Neglect the size of the pulley. figure — add your own
  18. 12‑212The roller at \(A\) is moving with \(v_A = 4\) m/s and has \(a_A = 2\) m/s² when \(x_A = 3\) m. Determine the velocity and acceleration of block \(B\) at this instant. figure — add your own
  19. 12‑213The man pulls the boy up to the tree limb \(C\) by walking backward at constant speed \(1.5\) m/s. Determine the speed at which the boy is being lifted at the instant \(x_A = 4\) m. Neglect the size of the limb. When \(x_A = 0, y_B = 8\) m, so \(A\) and \(B\) are coincident; the rope is \(16\) m long. figure — add your own
  20. 12‑214The man pulls the boy up to the tree limb \(C\) by walking backward. If he starts from rest when \(x_A = 0\) and moves backward with constant acceleration \(a_A = 0.2\) m/s², determine the speed of the boy at the instant \(y_B = 4\) m. Neglect the size of the limb. When \(x_A = 0, y_B = 8\) m, the rope is \(16\) m long. figure — add your own
  21. 12‑215The motor draws in the cord at \(B\) with \(a_B = 2\) m/s². When \(s_A = 1.5\) m, \(v_B = 6\) m/s. Determine the velocity and acceleration of the collar at this instant. figure — add your own
  22. 12‑216If block \(B\) is moving down with velocity \(v_B\) and acceleration \(a_B\), determine the velocity and acceleration of block \(A\) in terms of the parameters shown. figure — add your own
  23. 12‑217At the instant shown, the car at \(A\) is traveling at \(10\) m/s around the curve while increasing its speed at \(5\) m/s². The car at \(B\) is traveling at \(18.5\) m/s along the straightaway and increasing its speed at \(2\) m/s². Determine the relative velocity and relative acceleration of \(A\) with respect to \(B\) at this instant. figure — add your own
  24. 12‑218The boat can travel with speed \(16\) km/h in still water. The destination is located along the dashed line. If the water is moving at \(4\) km/h, determine the bearing angle \(\theta\) at which the boat must travel to stay on course. figure — add your own
  25. 12‑219The car is traveling at constant speed \(100\) km/h. If the rain is falling at \(6\) m/s in the direction shown, determine the velocity of the rain as seen by the driver. figure — add your own
  26. 12‑220Two planes \(A\) and \(B\) are flying at the same altitude. If their velocities are \(v_A = 500\) km/h and \(v_B = 700\) km/h such that the angle between their straight‑line courses is \(\theta = 60^\circ\), determine the velocity of plane \(B\) with respect to plane \(A\). figure — add your own