$A$ particle is projected with velocity $u$ from point $M$ on a frictionless incline of length $20\sqrt{2} \, m$. If it crosses a $40 \, m$ wide well at an angle of $45^o$,what should be its velocity at point $M$?

$A$ projectile of mass $m$ is fired with velocity $v$ from the point $P$ at an angle $45^{\circ}$ with the horizontal. The magnitude of change in momentum,when it passes through the point $Q$ on the same horizontal line on which $P$ lies,is :

For a given initial velocity,a projectile is fired at angles of $30^\circ$ and $60^\circ$. Find the ratio of their maximum heights.

$A$ player kicks a football at an angle of $30^{\circ}$ with the horizontal with an initial speed of $30 \,ms^{-1}$. $A$ second player, standing at a distance of $21 \sqrt{3} \,m$ from the first player in the direction of the kick, starts running to catch the ball at the same instant it is kicked. What is the minimum speed of the second player to catch the ball before it hits the ground? (Take acceleration due to gravity $g = 10 \,ms^{-2}$)

State whether the following statements are true or false:
$(a)$ The time of flight of a projectile depends only on the vertical component of the initial velocity.
$(b)$ The acceleration of a projectile at the maximum height is zero.
$(c)$ The maximum range of a projectile depends on the angle of projection.

$A$ point $P$ on the circumference of a wheel is initially in contact with the ground. If the radius of the wheel is $5 \, m$,what will be the displacement of point $P$ when the wheel rolls forward by half a revolution?