$A$ car is going round a circle of radius $R_1$ with constant speed. Another car is going round a circle of radius $R_2$ with constant speed. If both of them take the same time to complete the circles,the ratio of their angular speeds and linear speeds will be .........

The initial velocity of a projectile is $\vec{u} = (4\hat{i} + 3\hat{j})\,m/s$. It is moving with uniform acceleration $\vec{a} = (0.4\hat{i} + 0.3\hat{j})\,m/s^2$. The magnitude of its velocity after $10\,s$ is.........$m/s$.

$A$ cricketer can throw a ball to a maximum horizontal distance of $100\,m$. With the same speed,how high above the ground can the cricketer throw the same ball (in $,m$)?

$A$ ball is projected at an angle of $45^{\circ}$ with the horizontal. It passes through a wall of height $h$ at a horizontal distance $d_1$ from the point of projection and strikes the ground at a distance $d_1+d_2$ from the point of projection. Then $h$ is:

$A$ particle moves in space according to the equation $\vec{r} = (\sin t \hat{i} + \cos t \hat{j} + t \hat{k}) \text{ m}$. Find the time $t$ when the position vector and acceleration vector are perpendicular to each other.

In a circus,a performer throws an apple towards a hoop held at $45 \, m$ height by another performer standing on a high platform (see figure). The thrower aims for the hoop and throws the apple with a speed of $24 \, m/s$. At the exact moment that the thrower releases the apple,the other performer drops the hoop. The hoop falls straight down. At what height above the ground does the apple go through the hoop (in $, m$)?