$A$ particle is moving in a circle of radius $R$ with constant speed $V$. The magnitude of average acceleration after half revolution is

$A$ small boy is throwing a ball towards a wall $6 \,m$ in front of him. He releases the ball at a height of $1.4 \,m$ from the ground. The ball bounces from the wall at a height of $3 \,m$,rebounds from the ground and reaches the boy's hand exactly at the point of release. Assuming the two bounces (one from the wall and the other from the ground) to be perfectly elastic,how far ahead of the boy did the ball bounce from the ground (in $,m$)?

At a given instant of time, the position vector of a particle moving in a circle with a velocity $\vec{v} = 3 \hat{i} - 4 \hat{j} + 5 \hat{k}$ is $\vec{r} = \hat{i} + 9 \hat{j} - 8 \hat{k}$. Its angular velocity $\vec{\omega}$ at that time is:

$A$ merry-go-round starts from rest and accelerates at $0.4 \ rad \ s^{-2}$ for the first $5 \ s$. It then rotates at this constant angular velocity for $30 \ s$ and finally comes to rest with the same magnitude of angular deceleration. What is the total linear distance traveled by a child sitting $3 \ m$ from the center of the merry-go-round?

$A$ projectile can have the same range $(R)$ for two angles of projection. Their initial velocities are same. If $T_1$ and $T_2$ are times of flight in two cases,then the product of two times of flight is directly proportional to