Relative Motion of a Particle

Relative motion of a particle is considered when a particle moves with respect to a reference frame that itself moves relative to another frame treated as fixed. This description separates the motion into absolute, relative, and transport components.

Three types of motion

Absolute motion is particle motion relative to the fixed frame. Relative motion is motion relative to the moving frame. Transport motion is the motion of the point of the moving frame that instantaneously coincides with the particle.

Velocity addition

The absolute velocity of a particle equals the vector sum of its relative and transport velocities:

$$\vec v_a=\vec v_r+\vec v_e.$$

The relative velocity $\vec v_r$ describes particle motion with respect to the moving reference frame. The transport velocity $\vec v_e$ is the velocity of the point of the moving frame that instantaneously coincides with the particle.

This is a vector relation, so velocity magnitudes generally cannot simply be added algebraically; their directions must be taken into account.

Transport velocity

If the moving frame undergoes translation only, all its points have the same transport velocity. If it also rotates, then for a point with position vector $\vec r$ from a selected moving-frame origin $O'$, $\vec v_e=\vec v_{O'}+\vec\omega_e\times\vec r$.

Relative velocity

Relative velocity is measured by an observer moving with the moving coordinate frame. For example, for a slider moving in a slot fixed to a moving link, relative velocity is directed along the slot.

Vector addition

The equation $\vec v_a=\vec v_r+\vec v_e$ can be solved by resolving vectors into components or by constructing a velocity triangle. The convenient method depends on which directions and magnitudes are known.

Example

A person walks along a train car at 1.5 m/s relative to the car while the train moves along a straight track at 12 m/s in the same direction. The person’s absolute velocity is 13.5 m/s. If the person walks in the opposite direction, using the same sign convention the absolute velocity is 10.5 m/s.

Common mistakes

• confusing relative and transport velocities;
• adding velocity magnitudes without considering directions;
• for a rotating moving frame, using only the velocity of its origin as the transport velocity;
• failing to state the reference frame relative to which each velocity is defined.