$A$ body of mass $0.5 \,kg$ is projected under gravity with a speed of $98 \,m/s$ at an angle of $30^\circ$ with the horizontal. The change in momentum (in magnitude) of the body is ......... $N-s$.

The maximum range of a bullet fired from a toy pistol mounted on a car at rest is $R_0 = 10 \, m$. What will be the acute angle of inclination of the pistol for maximum range when the car is moving in the direction of firing with uniform velocity $v = 20 \, m/s$,on a horizontal surface? $(g = 10 \, m/s^2)$

$A$ horizontal plane supports a stationary vertical cylinder of radius $R = 1 \ m$ and a disc $A$ attached to the cylinder by a horizontal thread $AB$ of length $l_0 = 2 \ m$ (seen in figure,top view). An initial velocity $v_0 = 1 \ m/s$ is imparted to the disc as shown in the figure. How many seconds will it move along the plane until it strikes against the cylinder? (All surfaces are assumed to be smooth.)

From the ground level,a ball is to be shot with a certain speed. The graph shows the range $(R)$ of the particle versus the angle of projection from the horizontal $(\theta)$. The values of $\theta_1$ and $\theta_2$ are:

$A$ ball of mass $160 \, g$ is thrown up at an angle of $60^{\circ}$ to the horizontal at a speed of $10 \, m/s$. The angular momentum of the ball at the highest point of the trajectory with respect to the point from which the ball is thrown is nearly $\left(g=10 \, m/s^{2}\right)$ (in $kg \cdot m^{2}/s$).

The figure shows the $(x, t)$ and $(y, t)$ diagrams of a particle moving in $2$ dimensions. If the particle has a mass of $500 \, g$,find the force (magnitude and direction) acting on the particle.