Kinematics

Kinematics describes motion without asking what causes it. Learn displacement, velocity and acceleration, the constant-acceleration equations, motion graphs, projectiles and rotational kinematics.

Before asking why something moves, engineers describe how it moves: where it is, how fast, and how quickly its speed changes. Stopping distances, cam profiles, conveyor timing and robot trajectories are all kinematics problems.

For constant acceleration, four simple equations cover almost everything. For varying motion, calculus (derivatives and integrals) links position, velocity and acceleration.

An analogy

Kinematics is like a sports commentator describing a race — positions, speeds, overtakes — without discussing the athletes' muscles (that's dynamics).

Position, velocity, acceleration

Velocity is the rate of change of position; acceleration is the rate of change of velocity. Speed is the magnitude of velocity. Deceleration is just negative acceleration.

Motion graphs

The slope of a position–time graph is velocity; the slope of a velocity–time graph is acceleration; the area under a velocity–time graph is displacement. Reading graphs is often faster than algebra.

Projectile motion

Ignoring air resistance, horizontal velocity stays constant and vertical motion has constant acceleration −g. Treat the two directions independently and link them through time. On level ground, 45° gives maximum range.

Rotational kinematics

Rotation has the same equations with θ, ω and α. A point at radius r has tangential speed ωr, tangential acceleration αr and centripetal acceleration ω²r toward the centre — the load on rotating parts.

Relative motion and mechanisms

Mechanism analysis (slider-cranks, four-bars, cams) uses relative velocity and acceleration: v_B = v_A + ω × r_B/A. Instantaneous centres and vector loop equations give link velocities; this is the basis of engine and robot kinematics.

Formulas

• Constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as
• Calculus relations: v = ds/dt, a = dv/dt
• Projectile range (level ground): R = v² sin 2θ / g
• Rotation: ω = 2πN/60, v = ωr, a_t = αr, a_n = ω²r

Worked example: Car acceleration

Given: 0 → 100 km/h (27.8 m/s) in 8 s

Find: Acceleration and distance covered.

• a = 27.8/8 = 3.47 m/s²
• s = ½ × 3.47 × 8² = 111 m

Answer: 3.47 m/s², 111 m

Worked example: Projectile

Given: v = 20 m/s at 45°

Find: Range, maximum height and flight time.

• R = 20² × sin 90°/9.81 = 40.8 m
• H = 20² sin² 45°/(2 × 9.81) = 10.2 m
• T = 2 × 20 sin 45°/9.81 = 2.88 s

Answer: R = 40.8 m, H = 10.2 m, T = 2.88 s

Common mistakes

• Wrong: Using km/h directly in SI equations. — Right: Convert: divide km/h by 3.6 to get m/s.
• Wrong: Applying constant-acceleration equations when acceleration varies. — Right: Use calculus or piecewise segments for varying acceleration.

Applications

• Road safety: Stopping distances combine reaction distance (constant speed) and braking distance (v²/2a).
• Cam & mechanism design: Follower displacement, velocity and acceleration curves are designed to limit jerk and vibration.
• Robotics: Trajectory planners generate smooth position/velocity/acceleration profiles for each joint.

Key takeaways

• v = u + at, s = ut + ½at², v² = u² + 2as for constant a.
• Slopes and areas of motion graphs give v, a and s.
• Projectiles: independent horizontal and vertical motion.
• Rotation: v = ωr, a_n = ω²r.

Going further

• Jerk and motion profiles: Jerk (da/dt) causes vibration and discomfort; S-curve profiles in elevators, CNC and robots limit it.
• Coriolis acceleration: In rotating frames (a slider on a rotating arm), an extra 2ω × v_rel term appears — important in mechanisms and robot dynamics.
• Numerical integration: Real trajectories (with drag, varying thrust) are integrated numerically (Euler, Runge–Kutta), the basis of simulation tools.