Question

A car is travelling at $95km\,h{{r}^{-1}}$ is $210m$ behind a truck travelling at $75km\,h{{r}^{-1}}$. How long will it take the car to reach the truck?

Answer 1

Hint: As the car is moving behind a truck and needs to catch up with the truck, we will calculate the relative velocity. We know that the velocity is related to displacement and time so we use this relation in the relative motion of car and truck, substitute corresponding values to find time taken. Convert the units as required.

Formula used:
$v=\dfrac{d}{t}$

Complete solution:
Give, a car is travelling behind a truck. Speed of car is-
$\begin{align}
  & 95km\,h{{r}^{-1}}=95\times \dfrac{1000}{3600}m{{s}^{-1}} \\
 & \Rightarrow 95km\,h{{r}^{-1}}=26.4m{{s}^{-1}} \\
\end{align}$

Therefore, the speed of the car is $26.4m{{s}^{-1}}$. The speed of the truck is-
$\begin{align}
  & 75km\,h{{r}^{-1}}=75\times \dfrac{1000}{3600} \\
 & \Rightarrow 75km\,h{{r}^{-1}}=20.8m{{s}^{-1}} \\
\end{align}$
Therefore, the speed of the truck is $20.8m{{s}^{-1}}$.

The relative velocity of a body with respect to another is the difference of velocity between two bodies.

The relative velocity of the car will be-
$\begin{align}
  & {{v}_{r}}=26.4-20.8 \\
 & \Rightarrow {{v}_{r}}=5.6m{{s}^{-1}} \\
\end{align}$
Therefore, the velocity of car relative to the truck is $5.6m{{s}^{-1}}$

The car will have to cover $210m$ in order to catch up with the truck. We know that,
$v=\dfrac{d}{t}$
Here, $v$ is the velocity
$d$ is the distance
$t$ is the time taken

We substitute given values for relative motion between car and truck in the above equation to get,
$\begin{align}
  & v=\dfrac{d}{t} \\
 & \Rightarrow 5.6=\dfrac{210}{t} \\
 & \Rightarrow t=\dfrac{210}{5.6} \\
 & \therefore t=37.5s \\
\end{align}$

Therefore, it takes $37.5s$ for the car to reach the truck.

Note:
If bodies are moving in opposite directions the velocity of one is taken as positive and the other is taken as negative, hence the velocities will be added. If we change the frame of reference, the relative velocity also changes, for example- the relative velocity of a body inside the car will be zero relative to the car. We must convert all units in one system or else the answers will not be correct.