Question

A police van moving on a highway with a speed of 30 kilometers per hour fires a bullet at a thief’s car speeding away in the same direction with a speed of 192 kilometers per hour. If the muzzle speed of the bullet is 150 meters per second, with what speed does the bullet hit the thief’s car? (Note: Obtain the speed which is relevant for damaging the thief’s car).

Answer 1

Hint: In this question, we need to determine the speed of the bullet with which the hit hits the thief’s car that will damage the car as well. For this, we will use Newton's equation of motion and relative speed concept as all the components are in motion in this question.

Complete step by step answer:
The SI unit of the speed of a moving body is meters per second. So, we will first convert the two velocities given in the question in kilometers per hour to meters per second. To convert kilometers to meters, multiply the absolute value by 1000 and to convert hours to seconds, multiply the absolute value by 3600.
So, here 30 kilometers per hour can be converted into meters per second as:
$
{v_p} = 30{\text{ kmph }} \\
 = \dfrac{{30 \times 1000}}{{60 \times 60}}{\text{ m/s}} \\
 = \dfrac{{25}}{3}{\text{ m/s}} \\
 $
Similarly,
$
 {v_t} = 192{\text{ kmph }} \\
 = \dfrac{{192 \times 1000}}{{60 \times 60}}{\text{ m/s}} \\
 = \dfrac{{160}}{3}{\text{ m/s}} \\
 $
As the bullet is fired from the moving car of the police so, the resultant speed of the bullet will be the sum of the speed of the police’s car and the muzzle speed of the bullet. Here, the speed of the muzzle of the bullet is 150 meters per second.
So, the speed of the bullet is given as:
$
{v_b} = {v_p} + {v_m} \\
 = \dfrac{{25}}{3} + 150 \\
 = \dfrac{{25 + 450}}{3} \\
 = \dfrac{{475}}{3}{\text{ m/s}} \\
 $
Let the speed of the bullet with which it hits the thief’s car be $v$.
Now, the speed with the bullet will hit the thief’s car will be given as the relative speed between the moving bullet and the thief’s car. Moreover, the direction of movement of the bullet and the thief’s car is the same, so, the relative speed between the two will be their arithmetic difference. So,
$
v = {v_b} - {v_t} \\
= \dfrac{{475}}{3} - \dfrac{{160}}{3} \\
= \dfrac{{475 - 160}}{3} \\
= \dfrac{{315}}{3}{\text{ m/s}} \\
{\text{ = 105 m/s}} \\
 $
Hence, the speed of the bullet with which the hit hits the thief’s car that will damage the car as well is 105 meters per second.

Note:
Relative speed is the speed of the moving body when seen from a different frame of reference. Students should always convert all the units required (or given in the question) to similar ones and then substitute them in the desired equation.