Relative Motion

Relative motion is the concept that motion can be observed and described in different ways based on the frame of reference of the observer. For example, the speed of a car would be observed differently relative to a cyclist riding in the same direction and a stationary pedestrian. A frame of reference is the position from where an observation is being made, independent of any other motion.

Consider a train moving past a platform at a station. There is a passenger on the train and another person standing on the platform. The person on the platform would say that from their frame of reference, the train is moving. However, the passenger on the train would say that in their frame of reference, the train is stationary and that the person on the platform is moving. This concept can be difficult because we are so use to describing motion from the frame of reference of the Earth – the train is moving relative to the Earth and the person on the platform is stationary relative to the Earth!

Consider the diagram below where a red car is moving right at 5m/s and the blue car is moving left at 3m/s. There is also a stationary observer on the side of the road.

We can make the following statements about the different observations of relative motion:

  • The stationary observer would say the red car is moving right at 5m/s and the blue car is moving left 3m/s.
  • The red car would say the stationary observer is moving left at 5m/s
  • The blue car would say the stationary observer is moving right at 3m/s
  • The red car would say the blue car is moving left at 8m/s
  • The blue car would say the red car is moving right at 8m/s

All these statements are correct for the frame of reference of the observer.

Understanding the Correct Notation

Questions can be stated using velocities in different ways:

  • Velocities can all be simply stated relative to the ground. For example, vR may represent the velocity of the red car.
  • Velocities can be stated as being relative to some object or frame of reference. For example vRG could represent the velocity of the red car relative to the ground, and, vRB can represent the velocity of the red car relative to the blue car.

Calculating Relative Velocity

When calculating the relative velocity of an object (A) to some other object (B), or stated with the above notation: vAB, you carry out a vector subtraction:

vAB = vA – vB 

However, vectors can not be directly subtracted. You will need to add the negative of the second vector:

vAB = vA + (-vB)

This works for relative velocity problems in one and two dimensions. 

Example 1:

Calculate the velocity of the red car relative to the blue car in the diagram below:

vRB = vR – vB 

vRB = vR + (-vB)

As this is a vector velocity problem, we need to take note of direction. For this problem, let’s say right is positive. Therefore, vR = 12m/s and vB = -7m/s, and using: vRB = vR + (-vB)

vRB = 12 + (–7)

vRB = 19m/s (right)

Examples of two dimension relative velocity problems are in the next section.