Relative Velocity In Two Dimensions

The concept of relative velocity, introduced in section 3.7 for motion along a straight line, can be easily extended to include motion in a plane or in three dimensions. Suppose that two objects A and B are moving with velocities v$_A$ and v$_B$ (each with respect to some common frame of reference, say ground.). 

Then, velocity of object A relative to that of B is:

$\mathbf{v_{AB}}=\mathbf{v_A}+\mathbf{v_B}$

and similarly, the velocity of object B relative to that of A is:

$\mathbf{v_{BA}}=\mathbf{v_B}+\mathbf{v_A}$

Therefore, $\mathbf{v_{AB}}=-\mathbf{v_{BA}}$

and $|\mathbf{v_{AB}}|=|\mathbf{v_{BA}}|$

Example 1 

Rain is falling vertically with a speed of 35 ms$^{–1}$. A woman rides a bicycle with a speed of 12 m$s^{–1}$ in east to west direction. What is the direction in which she should hold her umbrella?

Answer 

In Fig. 1 v$_r$ represents the velocity of rain and v$_b$ , the velocity of the bicycle, the woman is riding. 

Both these velocities are with respect to the ground. Since the woman is riding a bicycle, the velocity of rain as experienced by her is the velocity of rain relative to the velocity of the bicycle she is riding. 

That is

$\mathbf{v_{rb}}=\mathbf{v_r}+\mathbf{v_b}$

This relative velocity vector as shown in Fig. 1 makes an angle θ with the vertical. It is given by

tan θ = $\frac{v_b}{v_r}=\frac{12}{35}$ = 0.343

Or, $\cong 19^0$ 

Fig.1

Therefore, the woman should hold her umbrella at an angle of about 19° with the vertical towards the west. 

Note carefully the difference between this Example and the Example 4.1. In Example 4.1, the boy experiences the resultant (vector sum) of two velocities while in this example, the woman experiences the velocity of rain relative to the bicycle (the vector difference of the two velocities).