$A$ particle of mass $M$ is moving in a circle of fixed radius $R$ in such a way that its centripetal acceleration at time $t$ is given by $n^2Rt^2$ where $n$ is a constant. The power delivered to the particle by the force acting on it,is

If at any point on the path of a projectile its velocity is $u$ at an inclination $\alpha$ to the horizontal,then after what time will it move at right angles to its former direction?

$A$ projectile is thrown in the upward direction making an angle of $60^{\circ}$ with the horizontal with a velocity of $140 \,ms^{-1}$. Then the time after which its velocity makes an angle $45^{\circ}$ with the horizontal is (Acceleration due to gravity, $g=10 \,ms^{-2}$) (in $\,s$)

Three point particles $P, Q, R$ move in a circle of radius $r$ with different but constant speeds. They start moving at $t = 0$ from their initial positions as shown in the figure. The angular velocities (in $rad/s$) of $P, Q,$ and $R$ are $5\pi, 2\pi,$ and $3\pi$ respectively, in the same sense. Find the time interval after which they are at the same angular position.

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

$A$ shot is fired from a point at a distance of $200 \ m$ from the foot of a tower $100 \ m$ high,such that it just passes over it. The direction of the shot with respect to the horizontal is (in $^{\circ}$)