Particle Kinematics

FE Exam Weight: Dynamics accounts for 9-14 questions (~10% of the exam). Kinematics (motion description) and kinetics (forces causing motion) are the two main branches.

Rectilinear Motion (Straight Line)

Fundamental Relationships

$v = \frac{ds}{dt} \qquad a = \frac{dv}{dt} = \frac{d^2s}{dt^2} \qquad v \, dv = a \, ds$

Constant Acceleration Equations

These are the most commonly used equations on the FE exam:

$v = v_0 + at$ $s = s_0 + v_0 t + \frac{1}{2}at^2$ $v^2 = v_0^2 + 2a(s - s_0)$ $s = s_0 + \frac{1}{2}(v_0 + v)t$

Gravity: For free fall, a = g = 9.81 m/s² (or 32.2 ft/s²) downward. Choose your sign convention carefully!

Variable Acceleration

When a = f(t): integrate a(t) to get v(t), then integrate v(t) to get s(t). When a = f(v): use v dv = a ds and separate variables. When a = f(s): use v dv = a(s) ds and integrate.

Projectile Motion

A projectile moves under gravity only (air resistance neglected):

Maximum height: H = (v₀ sin θ)² / (2g)

Range (level ground): R = v₀² sin(2θ) / g

Maximum range occurs at θ = 45°.

Time of flight (level ground): T = 2v₀ sin θ / g

Curvilinear Motion

Normal-Tangential (n-t) Components

For a particle moving along a curved path:

$a_t = \frac{dv}{dt} \quad (\text{tangential — changes speed})$

$a_n = \frac{v^2}{\rho} \quad (\text{normal — changes direction; always points toward center of curvature})$

$a = \sqrt{a_t^2 + a_n^2}$

where ρ is the radius of curvature.

For circular motion at constant speed: at = 0 and an = v²/r (centripetal acceleration).

Polar (r-θ) Components

$v_r = \dot{r} \qquad v_\theta = r\dot{\theta}$ $a_r = \ddot{r} - r\dot{\theta}^2 \qquad a_\theta = r\ddot{\theta} + 2\dot{r}\dot{\theta}$

Relative Motion

The velocity of B relative to A: $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A}$

Similarly for acceleration: $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A}$

Example: Car A travels north at 60 km/h and Car B travels east at 80 km/h. The velocity of B relative to A: vB/A = vB - vA → magnitude = √(60² + 80²) = 100 km/h, directed southeast relative to A.