A moving block having mass $m$ collides with another stationary block having a mass of $4m.$ The lighter block comes to rest after the collision. When the initial velocity of the lighter block is $v,$ then the value of the coefficient of restitution $(e)$ will be:
1. $0.5$
2. $0.25$
3. $0.8$
4. $0.4$

A body initially at rest and sliding along a frictionless track from a height $h$ (as shown in the figure) just completes a vertical circle of diameter $\mathrm{AB}= D.$ The height ${h}$ is equal to:
                    
1. ${3\over2}D$
2. $D$
3. ${7\over4}D$
4. ${5\over4}D$

Consider a drop of rainwater having a mass of $1~\text{gm}$ falling from a height of $1~\text{km}.$ It hits the ground with a speed of $50~\text{m/s}.$ Take $g$  as constant with a value $10~\text{m/s}^2.$ The work done by the
(i) gravitational force and the (ii) resistive force of air is:

Two identical balls $\mathrm{A}$ and $\mathrm{B}$ having velocities of $0.5~\text{m/s}$ and $-0.3~\text{m/s}$ respectively collide elastically in one dimension. The velocities of $\mathrm{B}$ and $\mathrm{A}$ after the collision respectively will be:
1. $-0.5 ~\text{m/s}~\text{and}~0.3~\text{m/s}$ 
2. $0.5 ~\text{m/s}~\text{and}~-0.3~\text{m/s}$
3. $-0.3 ~\text{m/s}~\text{and}~0.5~\text{m/s}$
4. $0.3 ~\text{m/s}~\text{and}~0.5~\text{m/s}$

A body of mass 1 kg begins to move under the action of a time dependent force $F = 2 t$ $\hat{i} + 3 t^{2}\ \hat{j}$ N, where $\hat{i}$ and $\hat{j}$ are unit vectors along X and Y axis, What power will be developed by the force at the time (t) ?

1. $\left(2 t^{2} + 4 t^{4}\right) W$         

2. $\left(2 t^{3} + 3 t^{4}\right) W$

3. $\left(2 t^{3} + 3 t^{5}\right) W$         

4. $\left(2 t + 3 t^{3}\right) W$

What is the minimum velocity with which a body of mass m must enter a vertical loop of radius R so that it can complete the loop?

1. $\sqrt{2\mathrm{gR}}$

2. $\sqrt{3gR}$

3. $\sqrt{5gR}$

4. $\sqrt{gR}$
 

A bullet of mass $10$ g moving horizontal with a velocity of $400$ m/s strikes a wood block of mass $2$ kg which is suspended by light inextensible string of length $5$ m. As a result, the centre of gravity of the block is found to rise a vertical distance of $10$ cm. The speed of the bullet after it emerges horizontally from the block will be:

Two identical balls $A$ and $B$ having velocities of $0.5~\text{m/s}$ and $-0.3~\text{m/s}$, respectively, collide elastically in one dimension. The velocities of $B$ and $A$ after the collision, respectively, will be:

What is the minimum velocity with which a body of mass $m$ must enter a vertical loop of radius $R$ so that it can complete the loop?
1. $\sqrt{2 g R}$
2. $\sqrt{3 g R}$
3. $\sqrt{5 g R}$
4. $\sqrt{ g R}$