Kinematics of Particles

Introduction

Kinematics is the study of motion without considering the forces or torques that cause it. In engineering mechanics, a body may be modeled as a particle when its dimensions and rotation are unimportant for the motion being analyzed. This model is appropriate for the motion of a vehicle’s center of mass, a projectile, a slider, or a point on a mechanism.

A complete kinematic description requires position, velocity, and acceleration as functions of time. The most suitable coordinate system depends on the geometry of the motion. This chapter develops Cartesian, normal–tangential, and polar descriptions and applies them to rectilinear, curvilinear, projectile, relative, and constrained motion.

Learning Objectives

After studying this chapter, you should be able to:

1. Define position, displacement, velocity, and acceleration.
2. Analyze rectilinear motion using time-dependent relationships.
3. Resolve velocity and acceleration into Cartesian components.
4. Use normal–tangential coordinates for motion along a curved path.
5. Use polar coordinates for radial and transverse motion.
6. Analyze projectile and relative motion.
7. Formulate and solve basic constrained-motion problems.
8. Select an appropriate coordinate system and assess modeling assumptions.

1. Fundamental Kinematic Quantities

Position and displacement

Relative to a fixed reference frame, the position of a particle is represented by the position vector r(t). In Cartesian coordinates,

where x, y, and z are coordinates measured in metres, and i, j, and k are fixed unit vectors. The displacement between times t₁ and t₂ is

Displacement is a vector and is not generally equal to the total distance traveled. Distance is the scalar length of the actual path.

Velocity

Average velocity is displacement divided by elapsed time:

Instantaneous velocity is the time derivative of position:

Velocity is tangent to the particle’s path. Its SI unit is metres per second (m/s).

Acceleration

Average acceleration is the change in velocity divided by elapsed time. Instantaneous acceleration is

Its SI unit is metres per second squared (m/s²). Acceleration may exist even when speed is constant, because the direction of velocity may change.

Speed

Speed is the magnitude of velocity:

The distinction between velocity and speed is essential: velocity includes direction, whereas speed does not.

2. Rectilinear Motion

Rectilinear motion occurs along a straight line. Let s denote position along the line. Then

and

The following chain-rule relationship is useful when acceleration is specified as a function of position or velocity:

Thus, the basic differential relationships are

and, when appropriate,

Signs must be assigned consistently. A positive direction is selected first; displacement, velocity, and acceleration in that direction are positive.

Constant acceleration

If acceleration is constant, integration gives

Here s₀ and v₀ are initial position and velocity, s and v are values at time t, and a is constant acceleration. These equations are not valid when acceleration varies unless a suitable average or piecewise model is justified.

Worked Example 1: Braking vehicle

A vehicle traveling at 25 m/s brakes with constant acceleration of −5 m/s². Determine stopping time and stopping distance.

At rest, v = 0. From v = v₀ + at,

0 = 25 − 5t, so t = 5 s.

Using v² = v₀² + 2a(s − s₀),

0 = 25² − 10(s − s₀). Therefore,

s − s₀ = 62.5 m.

The model assumes a straight path, constant braking acceleration, and no delay between braking command and deceleration.

3. Cartesian Description of Curvilinear Motion

For planar motion,

Differentiating with respect to time gives

and

where dots denote time derivatives. Hence,

and

The direction of velocity relative to the positive x-axis is

provided the quadrant is selected correctly. Cartesian coordinates are particularly effective when x(t) and y(t) are known independently or when acceleration components are constant.

Projectile motion

For a projectile near Earth, neglecting air resistance and Earth rotation, acceleration is approximately constant and downward:

where g ≈ 9.81 m/s². If the projectile is launched from (x₀,y₀) with speed v₀ at angle θ above the horizontal,

Horizontal velocity is constant in this idealization, while vertical velocity changes uniformly.

Worked Example 2: Projectile from a platform

A ball is launched at 20 m/s at 30° above the horizontal from a platform 10 m above level ground. Find the time to impact and horizontal range. Take g = 9.81 m/s².

The vertical equation is

0 = 10 + (20 sin30°)t − 4.905t².

Since 20 sin30° = 10 m/s, this becomes 4.905t² − 10t − 10 = 0. The positive root is

t = (10 + √(100 + 196.2))/9.81 ≈ 2.80 s.

The horizontal speed is 20 cos30° ≈ 17.32 m/s, so the range is

x − x₀ = 17.32(2.80) ≈ 48.5 m.

The negative quadratic root is rejected because it represents a time before launch.

4. Normal–Tangential Coordinates

When a particle follows a known curved path, it is often convenient to resolve acceleration into tangent and normal components. Let e_t be the unit tangent vector in the direction of motion and e_n point toward the instantaneous center of curvature. Then

and

The components are

where ρ is the radius of curvature. Therefore,

The tangential component changes speed; the normal component changes direction. For uniform-speed motion, a_t = 0, but a_n remains nonzero on a curved path. At a straight segment, ρ → ∞ and the normal component vanishes.

The radius of curvature for a planar path y(x) is

where the derivatives are evaluated at the point of interest. This expression applies where the curve is sufficiently smooth and d²y/dx² is not zero.

Application: vehicle cornering

For a vehicle traveling at speed v around a curve of radius ρ, the required lateral acceleration is v²/ρ. Increasing speed increases this requirement quadratically. Tire–road friction, suspension design, and passenger comfort therefore impose practical limits on speed, even when the vehicle’s speed is constant.

5. Polar Coordinates

Polar coordinates are useful when motion is naturally described by distance from a fixed origin and an angle. The position is

where r is radial distance and e_r is the outward radial unit vector. The transverse unit vector e_θ is perpendicular to e_r in the direction of increasing θ. Because these unit vectors rotate, their time derivatives are

Differentiating position gives velocity:

A second differentiation gives acceleration:

The radial acceleration is a_r = r̈ − rθ̇², and the transverse acceleration is a_θ = rθ̈ + 2ṙθ̇. The term −rθ̇² represents inward acceleration associated with angular motion; 2ṙθ̇ appears when radial and angular motions occur simultaneously.

For circular motion with constant radius and angular speed, ṙ = r̈ = θ̈ = 0. Therefore,

Since v = rθ̇, this agrees with a_n = v²/r.

6. Relative Motion

The motion of one particle can be described relative to another. For particles A and B observed in the same nonrotating reference frame,

where r_B/A is the position of B relative to A. Differentiation gives

and

These equations apply directly when the reference frames are parallel and nonrotating. In a translating but nonrotating frame, relative velocity and acceleration are still obtained by subtraction. If the observing frame rotates, additional transport terms arise and the simple equations are insufficient.

Relative motion is central to aircraft navigation, robotic tracking, conveyor systems, and collision analysis. The relative velocity is

If this vector is directed toward a target, the separation may decrease even when both particles are moving rapidly in the same general direction.

Worked Example 3: Crossing conveyor belts

A package moves east at 2 m/s on one conveyor while a second conveyor moves north at 1.5 m/s. The velocity of the second package relative to the first is

v_B/A = (0i + 1.5j) − (2i + 0j) = −2i + 1.5j m/s.

Its relative speed is

|v_B/A| = √(2² + 1.5²) = 2.50 m/s.

The direction is 36.9° north of west. This result is a velocity relative to the first package, not an absolute ground velocity.

7. Constrained Motion

A constraint imposes a geometric relationship between coordinates. If the coordinates satisfy

then differentiating once gives a velocity relationship:

A second differentiation gives the corresponding acceleration relationship. For a fixed-length cable connecting two translating points, a common constraint is

where L is constant. Differentiation gives

and therefore the ends have equal speed in opposite directions. This simple relation applies only when the cable is taut, inextensible, and routed over ideal fixed pulleys. Slack cables, elastic cables, moving pulleys, and pulley rotation require modified relations.

Illustrative cable system

Consider two blocks connected by a single inextensible cable over a fixed pulley. If block A moves downward with coordinate x and block B moves upward with coordinate y, the constant-length condition is x + y = L. Thus v_A = −v_B under the selected coordinate directions. If a movable pulley is attached to one block, the cable may contain two segments whose lengths change with the block motion; the resulting relation can instead be of the form x + 2y = L, leading to ẋ + 2ẏ = 0. The geometry, not memorization, determines the coefficient.

8. Choosing Coordinates and Checking Solutions

A sound solution follows a deliberate process:

1. Define the reference frame and positive directions.
2. Identify whether the path is straight, known-curved, or naturally radial.
3. Draw the position, velocity, and acceleration directions.
4. Write the geometric or kinematic equations before substituting values.
5. Apply initial conditions and retain physically meaningful roots.
6. Check units, signs, limiting behavior, and magnitude.

Cartesian coordinates are generally best for independent horizontal and vertical motion. Normal–tangential coordinates are best when the path and curvature are known. Polar coordinates are best for mechanisms with rotating arms, sliders, or radial motion.

Common errors include confusing distance with displacement, treating speed as a vector, omitting normal acceleration in curved motion, using constant-acceleration equations for variable acceleration, and assigning cable ratios without examining the cable geometry.

9. Applied Case: Robotic Pick-and-Place Motion

Background

Consider an illustrative industrial robot that moves a gripper from a supply point to a workstation. The gripper travels along a circular arc of radius 0.80 m while its speed increases uniformly from 0.40 m/s to 1.20 m/s over 2.0 s.

Problem

Determine the tangential acceleration and the normal acceleration at the end of the motion. Assess which component is associated with speed change and which is associated with path curvature.

Analysis and solution

Because speed changes uniformly,

a_t = (1.20 − 0.40)/2.0 = 0.40 m/s².

At the final speed, the normal acceleration is

a_n = v²/ρ = 1.20²/0.80 = 1.80 m/s².

The total acceleration magnitude is

a = √(a_t² + a_n²) = √(0.40² + 1.80²) ≈ 1.84 m/s².

The acceleration direction is mainly normal to the path because the normal component is substantially larger than the tangential component.

Outcome and learning points

The robot controller must coordinate speed commands with path curvature. At higher speed or smaller radius, normal acceleration increases rapidly. A controller that limits only tangential acceleration may still produce excessive lateral loads, vibration, or positioning error. Practical design should also account for actuator limits, structural flexibility, control sampling, payload variation, and emergency stopping distance.

10. Limitations and Professional Practice

Particle kinematics does not determine forces, stresses, power, or energy. Those require kinetics and material or rigid-body models. The particle approximation may be inappropriate when rotation, body deformation, size, or orientation affects performance. Environmental effects such as aerodynamic drag, rolling resistance, wind, and vibration may invalidate ideal projectile or constant-acceleration models.

In engineering software, numerical differentiation can amplify measurement noise. Position data should be sampled adequately and filtered with methods that do not introduce unacceptable phase delay. Units must be explicit, especially when converting between degrees and radians, kilometres per hour and metres per second, or millimetres and metres. Automated and AI-based motion analysis can assist with trajectory fitting and anomaly detection, but engineers remain responsible for validating coordinate definitions, assumptions, sensor quality, and safety consequences.

Chapter Summary

Kinematics describes particle motion through position, velocity, and acceleration without analyzing the forces that produce motion. Rectilinear motion is described by derivatives of position along a line, with constant-acceleration equations available only when acceleration is constant. Cartesian coordinates resolve motion into fixed orthogonal directions and are especially useful for projectile motion. Normal–tangential coordinates separate acceleration into a tangential component that changes speed and a normal component v²/ρ that changes direction. Polar coordinates describe radial and angular motion and include both radial and transverse acceleration terms. Relative-motion equations relate observations of two particles in a common nonrotating frame. Constrained motion follows from differentiating geometric length or position relationships. Correct coordinate selection, sign conventions, dimensional checks, and explicit assumptions are essential to reliable engineering analysis.

Key Terms

Particle: An idealized body whose dimensions and rotation are neglected.

Kinematics: The study of motion without considering its causes.

Position: Location of a particle relative to a reference frame.

Displacement: Vector change in position.

Distance: Scalar length of the path traveled.

Velocity: Time rate of change of position.

Speed: Magnitude of velocity.

Acceleration: Time rate of change of velocity.

Rectilinear motion: Motion along a straight line.

Curvilinear motion: Motion along a curved path.

Projectile motion: Motion under gravity with air resistance neglected.

Cartesian coordinates: Motion description using mutually perpendicular fixed axes.

Normal direction: Direction toward the instantaneous center of curvature.

Tangential direction: Direction tangent to the path.

Radius of curvature: Local radius of the path’s osculating circle.

Polar coordinates: Description using radial distance and angular position.

Relative motion: Motion of one particle measured with respect to another.

Constraint: Geometric or physical condition restricting motion.

Inextensible cable: Cable whose length remains constant.

Reference frame: Coordinate system and observer used to describe motion.

Review Questions

Short Answer Questions

1. Distinguish displacement from distance.
2. Why can a particle with constant speed have nonzero acceleration?
3. State the conditions required for the constant-acceleration equations.
4. What physical effects do tangential and normal acceleration represent?
5. Why are polar unit vectors time-dependent?

Long Answer Questions

1. Derive the Cartesian velocity and acceleration expressions for planar motion.
2. Explain projectile motion and derive its horizontal and vertical position equations.
3. Derive the normal–tangential acceleration relationship and interpret each term.
4. Derive the polar-coordinate velocity and acceleration equations.
5. Explain how an inextensible-cable constraint produces velocity and acceleration relationships.

Analytical / Application Questions

1. A car follows a 60 m radius curve at 18 m/s while its speed increases at 1.5 m/s². Determine its tangential, normal, and total acceleration.
2. A stone is launched from ground at 24 m/s and 40° above horizontal. Determine its idealized time of flight and horizontal range.
3. A slider has radial position r = 0.50 + 0.10t² m and angular position θ = 0.40t rad. Determine its velocity and acceleration at t = 2.0 s using polar coordinates.

Multiple Choice Questions with Answers

Question 1

Which quantity is the time derivative of position?

A. Displacement

B. Velocity

C. Acceleration

D. Speed

Correct Answer: B. Velocity

Question 2

A particle moves at constant speed around a circular path. Which statement is correct?

A. Its acceleration is zero

B. Its tangential acceleration is nonzero and normal acceleration is zero

C. Its normal acceleration is nonzero and directed toward the center

D. Its velocity is constant

Correct Answer: C. Its normal acceleration is nonzero and directed toward the center

Question 3

For constant acceleration, which equation does not require time?

A. v = v₀ + at

B. s = s₀ + v₀t + ½at²

C. v² = v₀² + 2a(s − s₀)

D. s − s₀ = ½(v₀ + v)t

Correct Answer: C. v² = v₀² + 2a(s − s₀)

Question 4

In ideal projectile motion, the horizontal acceleration is

A. g

B. −g

C. zero

D. dependent on launch speed

Correct Answer: C. zero

Question 5

The normal component of acceleration is

A. dv/dt

B. v²/ρ

C. ρ/v²

D. rθ̈

Correct Answer: B. v²/ρ

Question 6

In polar coordinates, the transverse velocity component is

A. ṙ

B. r̈

C. rθ̇

D. −rθ̇²

Correct Answer: C. rθ̇

Question 7

If particle B has velocity 8 m/s east and particle A has velocity 3 m/s east, the velocity of B relative to A is

A. 11 m/s east

B. 5 m/s east

C. 5 m/s west

D. 24 m/s east

Correct Answer: B. 5 m/s east

Question 8

For a taut inextensible cable with constraint x + y = L, differentiation gives

A. ẋ + ẏ = L

B. ẋ − ẏ = 0

C. ẋ + ẏ = 0

D. ẍ + ÿ = L

Correct Answer: C. ẋ + ẏ = 0

Question 9

A vehicle travels around a 100 m radius curve at 10 m/s. Its normal acceleration is

A. 0.01 m/s²

B. 0.1 m/s²

C. 1.0 m/s²

D. 10 m/s²

Correct Answer: B. 0.1 m/s²

Question 10

Which coordinate system is generally most suitable when a particle’s path and radius of curvature are known?

A. Normal–tangential

B. Only polar

C. Only Cartesian

D. None; coordinates are unnecessary

Correct Answer: A. Normal–tangential