Kinematics of a Particle
The geometry of motion — position, velocity, and acceleration — studied without yet asking what forces produce it. This chapter develops the language of particle motion along straight lines and curved paths, in several coordinate systems, and extends it to linked and relative motion.
Chapter Objectives
- Distinguish position, displacement, distance, velocity, speed, and acceleration physically and dimensionally.
- Select and apply differential or constant-acceleration relations for straight-line motion.
- Interpret slopes and signed areas on position, velocity, and acceleration graphs.
- Resolve curved-path motion in rectangular, normal–tangential, or cylindrical coordinates.
- Formulate dependent-motion constraints and relative-motion vector equations using declared coordinates.
- Verify computed motion using units, signs, reversals, boundary conditions, and admissible contact events.
Master Section 12.2 cold before anything else — every later section reuses a ds = v dv and the three constant-acceleration formulas. When you pick up a problem, your first move is to ask which of the four variables (a, v, s, t) you're given and which you need; that choice alone tells you which kinematic equation to reach for.
Introduction
Mechanics studies how bodies respond to forces — whether they stay at rest, move steadily, or accelerate. Engineering mechanics splits into statics (equilibrium) and dynamics (accelerated motion). This chapter belongs to dynamics, which itself divides into two parts: kinematics, the geometry of motion, and kinetics, the link between motion and the forces that cause it.
Mechanics
The branch of physical science concerned with the state of rest or motion of bodies under the action of forces.
Statics vs. Dynamics
Statics treats bodies in equilibrium — at rest or moving at constant velocity. Dynamics treats accelerated motion, where velocity changes with time.
Kinematics vs. Kinetics
Kinematics describes motion geometrically — position, velocity, acceleration — without reference to its cause. Kinetics analyzes the forces responsible for that motion.
Problem-Solving Sequence
- Read carefully and connect the physical situation to the theory.
- Sketch a diagram and tabulate the given data.
- Choose a coordinate system and apply the relevant principles in mathematical form.
- Solve algebraically first, then substitute consistent units; report no more significant figures than the data justify.
- Check whether the answer is physically reasonable.
"Kinematics vs. kinetics" trips up almost everyone on the first exam. Memory hook: kinematics is the map (where/how fast), kinetics is the engine (why). All of Chapter 12 is the map.
Rectilinear Kinematics: Continuous Motion
A particle has mass but negligible size, so any rotation is ignored and its motion is the motion of its mass center. Along a straight-line (rectilinear) path, position, velocity, and acceleration can all be handled as signed scalars, since the direction is fixed along the coordinate axis.
Position
Located by a single coordinate s measured from a fixed origin O. Its magnitude is the distance from O; its algebraic sign gives the direction along the axis.
Displacement
The change in position, \(\Delta s = s' - s\). It is a vector quantity — distinct from distance traveled, which is a positive scalar equal to the total path length covered.
Velocity
The time rate of change of position. Average velocity is \(v_{avg}=\Delta s/\Delta t\); the instantaneous velocity is its limit as \(\Delta t \to 0\). Its sign matches that of ds.
Speed and Average Speed
Speed is the magnitude of velocity. Average speed is the total distance traveled divided by elapsed time — always positive, and generally different from average velocity, which uses displacement.
Acceleration
The time rate of change of velocity. When speed decreases the particle decelerates, and acceleration points opposite to velocity. A particle can have zero velocity yet nonzero acceleration.
Constant Acceleration
When acceleration is constant, \(a = a_c\), the three differential relations integrate into the familiar formulas relating \(a_c\), \(v\), \(s\), and \(t\). These apply only for constant acceleration with initial conditions \(s=s_0,\ v=v_0\) at \(t=0\). A common case is free fall near the earth, where \(a_c \approx 9.81\ \text{m/s}^2\) downward.
Important Points
- Rectilinear kinematics is straight-line motion; vectors reduce to signed scalars.
- Speed is the magnitude of velocity; average speed uses distance, average velocity uses displacement.
- A slowing particle is decelerating; it may have acceleration while momentarily at rest.
- The relation \(a\,ds = v\,dv\) follows from \(a=dv/dt\) and \(v=ds/dt\) by eliminating dt.
- The constant-acceleration formulas have limited use — they fail when acceleration varies.
Before plugging into the constant-acceleration set, prove acceleration is constant. If a, v, or even just the wording gives you a function of t, s, or v, those three formulas are off the table — go back to v=ds/dt, a=dv/dt, or a ds=v dv and integrate. This single check prevents most lost marks in this chapter.
Rectilinear Kinematics: Erratic Motion
When motion is erratic — its position, velocity, or acceleration cannot be captured by a single continuous function — graphs become the natural tool. Because the variables are tied together by \(v = ds/dt\), \(a = dv/dt\), and \(a\,ds = v\,dv\), one graph can be built from another by reading slopes and areas.
Slope Relations
The slope of the s–t curve at any instant equals the velocity; the slope of the v–t curve equals the acceleration. Differentiation lowers the curve's degree by one.
Area Relations
The area under the a–t curve over an interval equals the change in velocity; the area under the v–t curve equals the displacement. Integration raises the curve's degree by one.
v–s and a–s Graphs
From an a–s graph, the area under the curve gives \(\tfrac12(v_1^2 - v_0^2)\). From a v–s graph, acceleration is found as \(a = v\,(dv/ds)\) — velocity times the slope of the v–s curve.
Graph problems are bookkeeping, not calculus. Going down the chain (s→v→a) you take slopes; going up (a→v→s) you take areas and must carry the initial value forward. Label every breakpoint with its (t, value) before you start — that's where errors creep in.
General Curvilinear Motion
When a particle follows a curved path, both the magnitude and direction of its position, velocity, and acceleration can change. Vector analysis captures this naturally, with position given by the position vector \(\mathbf{r}(t)\) measured from a fixed point.
Position Vector
\(\mathbf{r} = \mathbf{r}(t)\) locates the particle on the path; both its length and direction vary as the particle moves.
Displacement Vector
Over a small interval the particle moves to \(\mathbf{r}' = \mathbf{r} + \Delta\mathbf{r}\); the displacement \(\Delta\mathbf{r}\) is found by vector subtraction, \(\Delta\mathbf{r} = \mathbf{r}' - \mathbf{r}\).
Velocity is Tangent to the Path
As \(\Delta t \to 0\), \(\Delta\mathbf{r}\) aligns with the tangent, so velocity is always tangent to the path. Its magnitude (speed) equals \(ds/dt\), the rate of change of the path length s.
Acceleration is Tangent to the Hodograph
Acceleration tracks changes in both the magnitude and direction of velocity. Plotting the velocity vectors from a common origin traces the hodograph; acceleration is tangent to that curve and, in general, not tangent to the path — it points toward the concave side.
Important Points
- Curvilinear motion changes both the magnitude and direction of position, velocity, and acceleration.
- Velocity is always tangent to the path.
- Acceleration is tangent to the hodograph — generally not to the path.
One line to carry into every curved-motion problem: velocity is tangent to the path, acceleration is not. Acceleration always leans toward the inside of the curve because it must account for the turning of v, not just its speeding up.
Curvilinear Motion: Rectangular Components
When a path is conveniently described by x, y, z coordinates, the motion can be resolved into independent components along each fixed axis. Because the unit vectors i, j, k are constant in direction, differentiating the position vector only changes the component magnitudes.
Component Resolution
Position, velocity, and acceleration each split into x, y, z parts. Along each axis the motion is rectilinear, so \(v = ds/dt\), \(a = dv/dt\), and \(a\,ds = v\,dv\) apply to that component alone.
Dot Notation
An overdot denotes a time derivative: \(\dot x = v_x\), \(\ddot x = a_x\). Velocity components are first derivatives of the coordinates; acceleration components are second derivatives.
Path Equation and the Chain Rule
In two dimensions, a known path \(y = f(x)\) links the x and y velocity and acceleration components through the chain rule of calculus.
The power of rectangular components is that the axes don't talk to each other: x-motion is solved independently of y-motion. When a path \(y=f(x)\) is given, the chain rule is the bridge — practice \(\dot y = (dy/dx)\dot x\) and \(\ddot y = (d^2y/dx^2)\dot x^2 + (dy/dx)\ddot x\) until they're automatic. This is the engine of projectile motion next.
Motion of a Projectile
Free-flight projectile motion is a special case of curvilinear motion in rectangular components. With air resistance neglected, the only acceleration is gravity, constant and downward. The horizontal and vertical motions are completely independent and share only the time of flight.
When air resistance is neglected, the sole force on a projectile is its weight, producing a constant downward acceleration \(a_c = g \approx 9.81\ \text{m/s}^2\), independent of horizontal motion or altitude.
Horizontal Motion
With \(a_x = 0\), the horizontal velocity stays constant throughout the flight; horizontal position grows linearly with time.
Vertical Motion
With \(a_y = -g\), the vertical motion follows the constant-acceleration formulas. Only two of the three vertical equations are independent.
Solving Projectile Problems
- A projectile has at most three unknowns — one horizontal equation plus two independent vertical equations.
- Horizontal velocity is constant; the resultant velocity is always tangent to the path.
- Choose the three equations that involve only the wanted unknowns; skip any equation containing an unneeded velocity.
Time is the glue between the two directions — solve for the time of flight from whichever direction gives it cleanly, then carry it across. Nine times out of ten the vertical equation \(y = y_0 + (v_0)_y t - \tfrac12 g t^2\) gets you the time, and the horizontal \(x = x_0 + (v_0)_x t\) finishes the range.
Curvilinear Motion: Normal & Tangential Components
When the path itself is known, it is often easiest to use axes that move with the particle: the tangential axis t along the direction of motion and the normal axis n pointing toward the center of curvature. This naturally separates "speeding up" from "turning."
n–t Coordinate Frame
The t axis is tangent to the path in the direction of increasing s; the n axis is perpendicular, directed toward the center of curvature O′ on the concave side. Their origin rides with the particle.
Tangential Acceleration
The component \(a_t\) measures the rate of change of speed. It acts forward when speeding up, backward when slowing. The rectilinear relations \(a_t = \dot v\) and \(a_t\,ds = v\,dv\) apply along it.
Normal (Centripetal) Acceleration
The component \(a_n = v^2/ρ\) measures the rate of change of direction of velocity. It always points toward the center of curvature, never away.
Radius of Curvature
For a planar path \(y = f(x)\), the local radius of curvature ρ is obtained from the path's first and second derivatives.
Two Special Cases
- Straight-line motion: \(ρ \to \infty\), so \(a_n = 0\) and acceleration is purely tangential — pure change in speed.
- Constant-speed curve: \(a_t = 0\), so \(a = a_n = v^2/ρ\) — pure change in direction (centripetal).
Pick n–t whenever you already know the path and care about turning. The mental split is clean: \(a_t\) is "foot on the gas/brake," \(a_n = v^2/ρ\) is "the road bending." Watch your units on ρ — mixing mm and m in the curvature formula is the most common slip here.
Curvilinear Motion: Cylindrical Components
When the angular motion of a radial line describes the problem most naturally, polar coordinates are the right choice. The particle is located by a radial distance r and a transverse angle θ; adding a z coordinate extends this to full cylindrical components.
Polar Coordinates
Position is set by radial coordinate r (outward from a fixed origin) and transverse angle θ (counterclockwise from a reference line). Unit vectors \(\mathbf{u}_r\) and \(\mathbf{u}_θ\) point in the directions of increasing r and θ and are mutually perpendicular.
Angular Velocity and Acceleration
\(\dotθ = dθ/dt\) is the angular velocity (rad/s); \(\ddotθ = d^2θ/dt^2\) is the angular acceleration (rad/s²). These drive the transverse motion.
Time Derivatives
If r and θ are given as functions of time, derivatives follow directly. If only the path \(r = f(θ)\) is known, the chain rule links \(\dot r\) to \(\dotθ\) and \(\ddot r\) to \(\ddotθ\).
The terms \(-r\dotθ^2\) (centripetal) and \(2\dot r\dotθ\) (Coriolis) are where students lose points — they're easy to drop. Reach for polar coordinates when the data is angular (a rotating arm, a cam, a tracking camera). If the path is given as \(r=f(θ)\), you must chain-rule your way to \(\dot r\) and \(\ddot r\) — there's no shortcut.
Absolute Dependent Motion of Two Particles
When two bodies are linked by an inextensible cord over pulleys, the motion of one depends on the other. Writing position coordinates from fixed datums and relating them through the constant cord length yields the velocity and acceleration relationships by differentiation.
Dependent Motion
One particle's motion is fixed by another's through a cord wrapped over pulleys. Position coordinates measured from fixed datums, along each path of motion, are related by the total cord length.
Position-Coordinate Equation
Each coordinate starts at a fixed datum and runs along the particle's direction of motion. Segments wrapped over pulleys (constant length) are excluded; the remaining lengths sum to a constant.
Multiple Cords
For systems with two or more cords, an intermediate coordinate (e.g. at a movable pulley) links the two particle coordinates. A separate length equation is written for each cord, then combined.
The whole trick is counting how many cord segments support the movable pulley — that integer becomes the coefficient (the "2" in \(2s_B + \dots\)). Get the position equation right and the velocity/acceleration relations fall out by differentiation. Sign of the answer tells you direction: negative means opposite to the positive datum direction.
Relative-Motion of Two Particles Using Translating Axes
When a path is complicated, it can help to view one particle from a frame attached to — and translating with — another particle. The frame only translates (its axes never rotate), so relative position, velocity, and acceleration combine with the carrier's motion by simple vector addition.
Translating Reference Frame
A second set of axes is attached to a chosen particle A and allowed only to translate relative to the fixed frame. The motion of B observed from this frame is the relative motion of B with respect to A.
Relative-Position Vector
The absolute positions relate through \(\mathbf{r}_B = \mathbf{r}_A + \mathbf{r}_{B/A}\), where \(\mathbf{r}_{B/A}\) is the position of B as seen from A.
Vector Addition and Unknowns
Because the three vectors form a triangle, at most two unknowns (magnitudes or directions) can be found. Solve graphically with the laws of sines/cosines, or by resolving into Cartesian components.
The subscript "cancellation" trick keeps you honest: in \(\mathbf{v}_B = \mathbf{v}_A + \mathbf{v}_{B/A}\), the two A's cancel to leave B = B. Set up your fixed axes, attach the moving frame to the particle whose motion you know, then resolve into i–j components — that turns one vector equation into two scalar ones, exactly enough for two unknowns.
Integrated Applications and Engineering Judgment
Physical meaning
Velocity is tangent to the path and acceleration describes how that velocity vector changes. A tangential component changes speed, while a normal component changes direction; neither component identifies a force until a kinetics model is introduced.
Problem-solving strategy
- Confirm that the requested result concerns motion only; if forces or mass determine the motion, combine this chapter with a kinetics principle.
- Sketch the path, choose a fixed datum and positive directions, and record initial or boundary conditions before differentiating or integrating.
- Match coordinates to the data: rectangular for component functions and projectiles, normal–tangential for known curvature, polar for radial/angular data, or a cord equation for dependent motion.
- Select the differential relation whose independent variable matches the given function, then solve symbolically before inserting consistent units.
- Check reversals, vector directions, cord compatibility, and whether the computed point lies within the stated time or path interval.
Recognition strategy
The velocity is supplied directly as a time function and no forces appear, so this is rectilinear kinematics: integrate for position, differentiate for acceleration, and inspect the sign of velocity for reversal.
Given information
- Positive \(s\) points along the rail; \(s(0)=0\), \(v(t)=2.0+1.5t-0.30t^2\) \(\text{m/s}\), and the target time is 4.0 s.
- Unknowns are \(s(4)\), \(v(4)\), \(a(4)\), and whether \(v\) becomes zero on the stated interval.
Model and interpretation
Treat the carriage as a particle moving on a fixed straight axis. The prescribed function is kinematic data rather than a force law, and its differentiability permits \(a=dv/dt\) while \(ds=v\,dt\).
Verification
Units: Integrating m/s over seconds gives meters, while differentiating m/s with respect to seconds gives m/s²; every reported quantity is dimensionally consistent.
Signs and direction: Velocity remains positive through 4.0 s, so the positive displacement is consistent; negative acceleration means slowing at the endpoint, not motion in the negative direction.
Plausibility: The average velocity implied by the displacement is 3.40 m/s, between the endpoint values and compatible with a profile that first rises and then gently falls.
Engineering interpretation
The carriage is still advancing at 4.0 s but is decelerating. The predicted zero crossing at 6.09 s must not be extrapolated unless the programmed profile is known to remain valid beyond its specified interval.
Recognition strategy
The particle is in free flight with constant vertical gravity and zero horizontal acceleration. Rectangular components separate the motion, and the ground-height condition selects the physically admissible positive time.
Given information
- Take \(+x\) horizontal in the launch direction and \(+y\) upward; \((x_0,y_0)=(0,6.00\ \text{m})\), \(v_0=22.0\ \text{m/s}\), and \(\alpha=35.0^\circ\).
- The target event is first contact with level ground at \(y=0\); air drag and earth curvature are neglected.
Model and interpretation
Treat the tool as a particle after release. Kinematics gives \(a_x=0\), \(a_y=-g\); contact ends the free-flight model, so only a positive root before impact is admissible.
Verification
Units: Each position term has meters, the velocity components retain m/s, and the quadratic time result is in seconds after dividing compatible SI quantities.
Signs and direction: The impact vertical component is negative under the upward-positive convention, and the discarded negative root describes an extrapolated pre-launch crossing rather than the flight.
Plausibility: The impact speed exceeds 22.0 m/s because the tool loses 6.00 m of elevation; independently, \(v^2=v_0^2+2g(6.00)=601.7\ \text{m}^2/\text{s}^2\) gives 24.53 m/s.
Engineering interpretation
The ideal trajectory predicts the first ground intersection and impact vector, useful for defining a preliminary exclusion zone. Real clearance planning must include drag, launch uncertainty, bounce, terrain, and measurement tolerances.
Recognition strategy
Radius is specified as a function of angle while angular rate data are known, so polar kinematics and the chain rule avoid constructing rectangular position functions.
Given information
- Positive \(\mathbf e_r\) is outward and positive \(\mathbf e_\theta\) follows increasing \(\theta\); evaluate at \(\theta=1.20\ \text{rad}\).
- Known rates are \(\dot{\theta}=0.800\ \text{rad/s}\) and \(\ddot{\theta}=-0.150\ \text{rad/s}^2\); forces are neither given nor required.
Model and interpretation
The head is modeled as a particle in a fixed plane. Use \(\dot r=r_\theta\dot{\theta}\), \(\ddot r=r_{\theta\theta}\dot{\theta}^2+r_\theta\ddot{\theta}\), then retain both geometric terms in polar acceleration.
Verification
Units: Radians are dimensionless, so \(r\dot{\theta}\) has m/s and both \(r\dot{\theta}^2\) and \(2\dot r\dot{\theta}\) have m/s² as required.
Signs and direction: The negative radial acceleration points inward even though \(\dot r\) is outward; angular deceleration does not by itself determine the transverse sign because the transport term is positive.
Plausibility: The acceleration magnitude is \(\sqrt{2.561^2+0.828^2}=2.69\ \text{m/s}^2\), and the dominant inward value is consistent with a 3.776 m/s transverse speed at a 4.72 m radius.
Engineering interpretation
The inward acceleration is generated kinematically by the rotating basis, not by an assumption about force. Actuator and bearing loads require the subsequent kinetics model, including head mass, drive forces, and structural flexibility.
Chapter recap
- Classify the given dependence before choosing a differential kinematics relation.
- Use coordinates aligned with the path or measurement data and keep velocity and acceleration as vectors.
- Treat constraints as geometry, differentiate them consistently, and test roots and directions for admissibility.
Before moving on
You should be able to integrate or differentiate motion data, resolve projectile and curvilinear motion, formulate cord and relative-motion constraints, and audit units, signs, reversals, and contact events.