Question

A police van moving on a highway with a speed of $30\dfrac{km}{h}$ fires a bullet at a thief’s car speeding away in the same direction with a speed of $192\dfrac{km}{h}$. If the muzzle speed of the bullet is $150\dfrac{km}{h}$, with what speed does the bullet hit the thief’s car?

A.)$105\dfrac{m}{s}$
B.)$205\dfrac{m}{s}$
C.)$210\dfrac{m}{s}$
D.)$250\dfrac{m}{s}$

Answer 1

Hint: There are some conditions where one or more objects are moving in a frame which is non-stationary with respect to an observer. In these cases, we apply the concept of Relative velocity.

Formula used:

$\overrightarrow{V}=\overrightarrow{{{V}_{AB}}}+\overrightarrow{{{V}_{BC}}}$

Complete step by step answer:
The relative velocity is defined as the velocity of an object or observer in the rest frame of another object or observer.

$\overrightarrow{{{V}_{AC}}}=\overrightarrow{{{V}_{AB}}}+\overrightarrow{{{V}_{BC}}}$
Where ${{V}_{AB}}$ is the velocity of body A with respect to body, ${{V}_{BC}}$ is the velocity of body B with respect to body C and ${{V}_{AC}}$ is the velocity of body A with respect to body C.

Muzzle velocity is the velocity by which a bullet leaves the shell or muzzle of a gun.

Police van is chasing a thief's car moving with $192\dfrac{km}{h}$ , with a speed of $30\dfrac{km}{h}$ on a highway.

Velocity of police van, say ${{\upsilon }_{p}}=30\dfrac{km}{h}=\dfrac{25}{3}\dfrac{m}{s}$
Velocity of thief’s car, say ${{\upsilon }_{t}}=190\dfrac{km}{h}=\dfrac{160}{3}\dfrac{m}{s}$

Velocity of bullet which police fires on the thief’s car, say ${{\upsilon }_{b}}=150\dfrac{m}{s}$
Final velocity of the bullet muzzle from police gun $={{\upsilon }_{b}}+{{\upsilon }_{P}}=150+\dfrac{25}{3}=\dfrac{475}{3}\dfrac{m}{s}$

Now let’s assume that the velocity of thief is zero and bullet’s motion can be seen with respect to the frame of thief at rest

${{\upsilon }_{\dfrac{b}{t}}}={{\upsilon }_{b}}-{{\upsilon }_{t}}=\dfrac{475}{3}-\dfrac{160}{3}=\dfrac{315}{3}=105\dfrac{m}{s}$
The speed by which the bullet hits the thief’s car is$105\dfrac{m}{s}$.

Hence, the correct option is A.

Note:
One of the very basic life examples of our encounter with relative velocity is that we are sitting in a train and we see another train moving off and feel we are moving even though we are stopped at the platform. This type of illusion occurs because there is no way to distinguish between the uniform motion and being stationary.