$A$ police jeep is chasing a thief's jeep. The police jeep is moving with a velocity of $45 \, km/h$ and the thief's jeep is moving with a velocity of $153 \, km/h$. The police fire a bullet with a muzzle velocity of $180 \, m/s$. What is the velocity with which the bullet strikes the thief's car in $m/s$?

$A$ man standing on a road holds his umbrella at $30^\circ$ with the vertical to keep the rain away. He throws the umbrella and starts running at $10 \, km/h$. He finds that raindrops are hitting his head vertically. The speed of raindrops with respect to the moving man will be:

$A$ person aiming to reach the exactly opposite point on the bank of a stream is swimming with a speed of $0.5\, m/s$ at an angle of $120^\circ$ with the direction of flow of water. The speed of water in the stream is..........$m/s$.

$A$ man walking briskly in rain with speed $v$ must slant his umbrella forward making an angle $\theta$ with the vertical. $A$ student derives the following relation between $\theta$ and $v: \tan \theta = v$ and checks that the relation has a correct limit: as $v \rightarrow 0, \theta \rightarrow 0,$ as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct? If not,guess the correct relation.

$A$ boy runs on a horizontal road with a speed of $4 \ m \ s^{-1}$ while it is raining. He sees that the rain is making an angle $\theta$ with the vertical while running from west to east. However,when he runs from east to west,the angle is $\alpha$. The rain is pouring down at an angle $45^{\circ}$ with the vertical and at a speed of $8 \ m \ s^{-1}$ as shown in the figure. The ratio $\frac{\tan \theta}{\tan \alpha}$ is,

Two particles $P_1$ and $P_2$ are moving with velocities $v_1$ and $v_2$ respectively. Which of the following statements about their relative velocity $v_{12}$ is true?