Dynamics: Kinematics, Newton's Laws, and Relative Motion

Separate kinematics from kinetics

Kinematics describes position, velocity, and acceleration without asking what caused the motion. Kinetics adds force and mass through Newton's laws or moment equations. On the FE Mechanical exam, deciding which side of that boundary you are on saves time. A projectile height question is usually kinematics. A block on an incline with friction is kinetics. A rotating disk with angular acceleration may require both.

Start by naming the body and model: particle, translating rigid body, fixed-axis rotation, or general plane motion. Then choose coordinates. Rectangular coordinates work well for straight-line motion and projectiles. Normal-tangential coordinates work well for curved paths because acceleration splits into tangential acceleration dv/dt and normal acceleration v^2/r toward the center of curvature.

Kinematics equations and limits

The constant-acceleration equations are fast but conditional. Use them only when acceleration is constant over the interval. If acceleration is a function of time, integrate a(t) to get v(t), then integrate v(t) to get position. If acceleration depends on position, use v dv/ds = a.

Use g = 9.81 m/s^2 or 32.2 ft/s^2 unless a problem gives a different value. Do not mix speed in mph with acceleration in ft/s^2 without converting.

Newton's second law workflow

For kinetics, draw a free-body diagram just as carefully as in statics. Then write sum F = ma in each active direction. The acceleration direction is not necessarily the same as the velocity direction. A car cresting a hill may have velocity tangent to the road while normal acceleration points toward the center of curvature.

In USCS, be careful with mass. Newton's law needs mass, not weight. If weight W is in lbf, mass is W/g in slugs when using ft/s^2. In SI, kg is already mass and weight is mg in newtons. Many wrong answers differ by a factor of 32.2 because weight was used directly as mass in USCS.

Relative motion and frames

Relative velocity and acceleration equations are vector equations. For translating frames, v_B = v_A + v_B/A and a_B = a_A + a_B/A. For rotating frames or rigid bodies, the angular terms add complexity, so define the reference point, angular velocity direction, and position vector before touching numbers.

For simple FE relative-motion questions, draw a velocity triangle. Label the known ground speeds and directions, then solve components. For two vehicles, wind and aircraft, or sliding collars, the biggest risk is using a scalar speed where a vector component is required.

Exam pacing check

Dynamics can consume time because the diagrams are less familiar than statics. If you do not know the model after 30 seconds, ask: Is acceleration given or requested? Is force given or requested? Is there rotation? Is motion along a curve? Those four questions usually point to kinematics, Newton's second law, rotation, or normal-tangential coordinates.