Relative Motion

Galilean transformations of position and velocity between reference frames in relative motion.

General idea of the topic

Velocity, position and trajectory depend on the observer. A passenger is at rest relative to the train and moves relative to the platform. To pass from one observer to another, velocities are added or subtracted according to the Galilean transformation, valid when the velocities are much smaller than the speed of light.

The topic defines reference frames, derives the relative velocity between two moving bodies or between a body and a medium such as air or water, and applies it to pursuit and interception problems.

What you will truly understand

By the end you will be able to change a problem from one reference frame to another, compute the relative velocity of two bodies and solve when and where they meet or cross.

You will also be able to choose the reference frame that simplifies the problem.

Recommended learning route

1. Reference Frames: what a reference frame is, inertial observers and coordinate transformation.
2. Relative Velocity: composition of velocities, in which the velocity of A relative to C is the sum of the velocity of A relative to B and that of B relative to C.
3. Pursuit and Interception Problems: meetings and catch-ups: conditions of equal position at the same instant.

Physical criterion you must acquire

The essential criterion is to write the velocities with the notation "of A relative to B" and check that the subscripts chain together. For pursuits it is useful to switch to a frame attached to one of the bodies, where the other moves with the relative velocity; the meeting occurs when the relative position vanishes.

Connection with real problems

• navigation with currents and wind.
• air traffic control.
• overtaking on roads.
• interception of objects.

Final goal of the topic

The final goal is to relate the motions seen by different observers and solve meeting problems by choosing the most convenient reference frame.