$A$ particle moves once around a circle of radius $1 \,m$. The time taken is $10 \,s$. The average velocity of the motion is

$A$ ball is released from rest from point $P$ of a smooth semi-spherical vessel as shown in the figure. The ratio of the centripetal force and normal reaction on the ball at point $Q$ is $A$,while the angular position of point $Q$ is $\alpha$ with respect to point $P$. Which of the following graphs represents the correct relation between $A$ and $\alpha$ when the ball goes from $Q$ to $R$?

The range of a projectile of mass $1\, kg$ is $80\, m$ while it attains a maximum height of $20\, m$. Its change in momentum during the complete motion is ............ $N-s$.

$A$ projectile crosses two walls of equal height $H$ symmetrically as shown. If the horizontal distance between the two walls is $d = 120\, m$ and the times at which the projectile crosses the walls are $t_1 = 2\, s$ and $t_2 = 6\, s$,then the range of the projectile is ........ $m$.

The average acceleration vector for a particle having a uniform circular motion over one complete revolution is

$A$ particle moves from the point $(2.0\hat i + 4.0\hat j) \, m$ at $t = 0$ with an initial velocity $(5.0\hat i + 4.0\hat j) \, m/s$. It is acted upon by a constant force which produces a constant acceleration $(4.0\hat i + 4.0\hat j) \, m/s^2$. What is the distance of the particle from the origin at time $t = 2 \, s$?