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Relative Velocity in One and Two Dimensions

This topic is part of the HSC Physics course under the section Motion on a Plane

HSC Physics Syllabus

  • calculate the relative velocity of two objects moving along the same line using vector analysis 
  • describe and analyse, using vector analysis, the relative positions and motions of one object relative to another object on a plane (ACSPH061) 
  • analyse the relative motion of objects in two dimensions in a variety of situations, for example: 
– a boat on a flowing river relative to the bank 
– two moving cars 
– an aeroplane in a crosswind relative to the ground (ACSPH060, ACSPH132) 

How to Find Relative Velocity?

 

Relative Velocity in One Dimension

Relative velocity is the velocity of an object with respect to another object or reference frame. In one-dimensional scenarios, this can be visualised as objects moving along a straight line. The relative velocity is given by the difference between the velocity of the object and the velocity of the reference frame.

$$v_{relative}=v_{object}-v_{reference}$$

Let's consider two objects, A and B, moving along the same straight line, where A has a velocity of `v_A` and B has a velocity of `v_B`. The relative velocity of B with respect to A is given by the difference between the velocities of B and A:

$$v_{BA}=v_B-v_A$$

 

 

For example, if A is moving to the right with a velocity of 5 m/s and B is moving to the right with a velocity of 3 m/s, then the relative velocity of B with respect to A is:

$$v_{BA}=3-5=-2 m s^{-1}$$

The negative sign indicates that B is moving to the left at 2 m/s relative to A. From the perspective of A, B appears to be moving to the left. 

 

 

If two objects are moving in different directions, the relative velocity is still the difference between their velocities.

If A is moving to the right with a velocity of 5 m/s and B is moving to the left with a velocity of 3 m/s, then the relative velocity of B with respect to A is:

$$v_{BA}=-3 - 5= -8 m s^{-1}$$

The positive sign indicates that B is to the left at 8 m/s relative to A. From the perspective A, B appears to be moving to the left at much greater speed than 3 m/s.

Relative Velocity in Two Dimensions

In two-dimensional scenarios, relative velocity becomes a bit more complex. We need to consider the velocities of the objects in both the x- and y-directions.

 

relative velocity
 

Let's consider two cars, A and B. Car A is moving north at 10 m/s while car B is moving to the east at 10 m/s.

The velocity of car B relative to car A is given by:

$$v_{BA}=v_B - v_A$$

However, we cannot subtract the magnitudes directly because they are in different directions. Instead, we should draw a diagram to visualise the magnitude and direction of the relative velocity vector.

 

The difference between `v_B` and `v_A` is equivalent to the sum of `v_B` and `-v_A`. Therefore, the relative velocity vector can be formed by finding the resultant vector between `v_B` and `-v_A`.

The magnitude of the relative velocity is calculated using Pythagorean theorem:

$$v_{relative}^2=10^2+10^2$$

$$v_{relative}=\sqrt{10^2+10^2}$$

$$v_{relative}=14 \hspace{1mm} m s^{-1}$$ 

The direction of the relative velocity is calculated using trigonometry:

$$\tan{\theta}=\frac{10}{10}$$

$$\theta = 45º$$

Therefore the velocity of car B relative to car A is 14 m/s 45 degrees above the x-axis or south east.

What about the velocity of car A relative to car B?

The velocity of car a relative to car B is given by:

$$v_{AB}=v_A - v_B$$

This can be visualised as the sum of vectors `v_A` and `-v_B`:

 

The magnitude of the relative velocity is calculated using Pythagorean theorem:

$$v_{relative}^2=10^2+10^2$$

$$v_{relative}=\sqrt{10^2+10^2}$$

$$v_{relative}=14 \hspace{1mm} m s^{-1}$$ 

The direction of the relative velocity is calculated using trigonometry:

$$\tan{\theta}=\frac{10}{10}$$

$$\theta = 45º$$

Therefore the velocity of car A relative to car B is 14 m/s 45 degrees above the x-axis or north east. In other words, the velocity of car A relative to car B has the same magnitude as the velocity of car B relative to A but a different direction.