Kinematics of a Particle

What is Kinematics?

Kinematics describes motion without considering forces. We study position, velocity, and acceleration as functions of time.

Rectilinear Motion (Straight Line)

Definitions

• Position: s(t) — location along a line
• Velocity: v = ds/dt (rate of change of position)
• Acceleration: a = dv/dt = d²s/dt²

Constant Acceleration Equations

| Equation | Variables | |----------|----------| | v = v₀ + at | v, v₀, a, t | | s = s₀ + v₀t + ½at² | s, s₀, v₀, a, t | | v² = v₀² + 2a(s-s₀) | v, v₀, a, s |

Projectile Motion (free fall only)

a_x = 0 → v_x = v₀ cos θ (constant) a_y = -g → v_y = v₀ sin θ - gt

x = v₀ cos θ × t y = v₀ sin θ × t - ½gt²

Range: R = v₀² sin(2θ) / g (flat ground, same elevation) Max height: H = (v₀ sin θ)² / 2g

Curvilinear Motion

Normal-Tangential Components (n-t)

For a particle on a curved path:

• a_t = dv/dt (tangential, changes speed)
• a_n = v²/ρ (normal, changes direction, toward center)

ρ = radius of curvature

Polar Coordinates (r-θ)

• v_r = ṙ (radial velocity)
• v_θ = rθ̇ (transverse velocity)
• a_r = r̈ - rθ̇²
• a_θ = rθ̈ + 2ṙθ̇

Example: Projectile

A ball is thrown at 20 m/s at 30° above horizontal. Find max height and range.

v₀x = 20 cos30° = 17.32 m/s
v₀y = 20 sin30° = 10 m/s

Max height: H = (10)²/(2×9.81) = 5.1 m
Time of flight: t = 2×10/9.81 = 2.04 s
Range: R = 17.32 × 2.04 = 35.3 m

Summary

Kinematics gives you position, velocity, and acceleration. Master the constant-acceleration equations and projectile motion first, then move to normal-tangential components.