Relative Motion

Relative motion describes how the position and velocity of one object appear from the perspective of another moving object. This means the velocity you observe depends on your own frame of reference.

The Concept

Imagine you are on a boat moving across a river, and you throw a ball forward. The speed and direction of the ball you see will depend on both the speed of the boat and the speed at which you throw the ball. To analyze this, we use the concept of relative velocity.

Relative Velocity Formula

\[ \vec{v}_{A/B} = \vec{v}_A - \vec{v}_B \]

Where:

\( \vec{v}_{A/B} \) is the velocity of object A relative to object B
\( \vec{v}_A \) is the velocity of object A with respect to the ground (fixed frame)
\( \vec{v}_B \) is the velocity of object B with respect to the ground

The subtraction is vector subtraction, meaning magnitude and direction matter.

Interactive Visualization

Adjust the sliders below to change the velocities and directions of two objects. See how the relative velocity vector changes accordingly.

Object A Velocity (m/s): 50

Object A Direction (°): 0

Object B Velocity (m/s): 30

Object B Direction (°): 90

Color legend:

Blue - Object A Velocity \( \vec{v}_A \)
Red - Object B Velocity \( \vec{v}_B \)
Green - Relative Velocity of A with respect to B \( \vec{v}_{A/B} \)

Watch this video for a visual explanation of relative motion:

Example Problem

A swimmer is swimming at 2 m/s upstream in a river flowing at 3 m/s downstream. What is the swimmer's velocity relative to the ground??

Step 1: Define vectors (choose downstream as positive x-direction):

Velocity of swimmer relative to water, \( \vec{v}_{S/W} = -2\, \text{m/s} \) (upstream = negative direction)
Velocity of water relative to ground, \( \vec{v}_{W/G} = +3\, \text{m/s} \) (downstream = positive)

Step 2: Find velocity of swimmer relative to ground:

\[ \vec{v}_{S/G} = \vec{v}_{S/W} + \vec{v}_{W/G} \]

Step 3: Substitute values:

\[ \vec{v}_{S/G} = -2 + 3 = +1\, \text{m/s} \]

The swimmer moves upstream relative to the water, but relative to the ground, he is still moving downstream at 1 m/s.

Summary

Relative velocity depends on the frame of reference.
Use vector subtraction to find the velocity of one object relative to another.
Directions and magnitudes both matter — consider vector components carefully.
Real-world examples include boats crossing rivers, cars passing each other, and airplanes in wind.