3.7 Relative velocity

RELATIVE VELOCITY

You must be familiar with the experience of travelling in a train and being overtaken by another train moving in the same direction as you are. While that train must be travelling faster than you to be able to pass you, it does seem slower to you than it would be to someone standing on the ground and watching both the trains. In case both the trains have the same velocity with respect to the ground, then to you, the other train would seem to be not moving at all. To understand such observations, we now introduce the concept of relative velocity.

Consider two objects \(A\) and \(B\) moving uniformly with average velocities \(v_{A}\) and \(v_{B}\) in one dimension, say along \(x\)-axis. If \(x_{A}(0)\) and \(x_{B}(0)\) are positions of objects \(A\) and \(B\), respectively at time \(t=0\), their positions \(x_{A}(t)\) and \(x_{B}(t)\) at time \(t\) are given by:
\(
\begin{array}{ll}
x_{A}(t)=x_{A}(0)+v_{A} t \\
x_{B}(t)=x_{B}(0)+v_{B} t 
\end{array}
\)
Then, the displacement from object \(A\) to object \(B\) is given by
\(
\begin{aligned}
&x_{B A}(t)=x_{B}(t)-x_{A}(t) \\
&=\left[x_{B}(0)-x_{A}(0)\right]+\left(v_{B}-v_{A}\right) t .
\end{aligned}
\)
Above equations, interpretations tell us that as seen from object \(A\), object \(B\) has a velocity \(v_{B}-v_{A}\) because the displacement from \(A\) to \(B\) changes steadily by the amount \(v_{B}-v_{A}\) in each unit of time. We say that the velocity of object \(B\) relative to object \(A\) is \(v_{B}-v_{A}\):
\(
v_{B A}=v_{B}-v_{A}
\)
Similarly, the velocity of the object \(A\) relative to object \(B\) is:
\(
v_{A B}=v_{A}-v_{B}
\). This shows: \(v_{B A}=-v_{A B}\)

Example 3.6:

Two parallel rail tracks run north-south. Train A moves north with a speed of \(54 \mathrm{~km} \mathrm{~h}^{-1}\), and train B moves south with a speed of \(90 \mathrm{~km} \mathrm{~h}^{-1}\). What is the
(a) velocity of \(\mathrm{B}\) with respect to \(\mathrm{A}\)?,
(b) velocity of ground with respect to B? and
(c) velocity of a monkey running on the roof of the train A against its motion (with a velocity of \(18 \mathrm{~km} \mathrm{~h}^{-1}\) with respect to the train A) as observed by a man standing on the ground?

Answer: Choose the positive direction of \(x\)-axis to be from the south to north. Then,

\(
\begin{aligned}
&v_{\mathrm{A}}=+54 \mathrm{~km} \mathrm{~h}^{-1}=15 \mathrm{~m} \mathrm{~s} \\
&v_{\mathrm{B}}=-90 \mathrm{~km} \mathrm{~h}^{-1}=-25 \mathrm{~m} \mathrm{~s}^{-1}
\end{aligned}
\)
Relative velocity of \(B\) with respect to \(A=v_{\mathrm{B}}-v_{\mathrm{A}}=\) \(-40 \mathrm{~m} \mathrm{~s}^{-1}\), i.e. the train \(B\) appears to \(A\) to move with a speed of \(40 \mathrm{~m} \mathrm{~s}^{-1}\) from north to south.
Relative velocity of ground with respect to \(B=0-v_{\mathrm{B}}=25 \mathrm{~m} \mathrm{~s}^{-1} \text {. }
\)
In (c), let the velocity of the monkey with respect to the ground be \(v_{\mathrm{M}}\). The relative velocity of the monkey with respect to \(A\), \(v_{M A}=v_{M}-v_{\mathrm{A}}=-18 \mathrm{~km} \mathrm{~h}^{-1}=-5 \mathrm{~m} \mathrm{~s}^{-1}\). Therefore, \(v_{\mathrm{M}}=(15-5) \mathrm{m} \mathrm{s}^{-1}=10 \mathrm{~m} \mathrm{~s}^{-1}\).