A particle is released from a height of S above the surface of the earth. At a certain height, its kinetic energy is three times its potential energy. The distance from the earth's surface and the speed of the particle at that instant are respectively: 

1. $\frac{\mathrm{S}}{2}, \frac{\sqrt{3\mathrm{gS}}}{2}$

2. $\frac{\mathrm{S}}{4}, \sqrt{\frac{3\mathrm{gS}}{2}}$

3. $\frac{\mathrm{S}}{4}, \frac{3\mathrm{gS}}{2}$

4. $\frac{\mathrm{S}}{4}, \frac{\sqrt{3\mathrm{gS}}}{2}$

Water falls from a height of 60 m at the rate of 15 kg/s to operate a turbine. The losses due to frictional force are 10% of the input energy. How much power is generated by the turbine? $\left(g=10 m/s^2\right)$

1. 12.3 kW 

2. 7.0 kW 

3. 10.2 kW 

4. 8.1 kW

Body A of mass 4m moving with speed u collides with another body B of mass 2m at rest. The collision is head-on and elastic in nature. After the collision, the fraction of energy lost by the colliding body A is:

1. $\frac{5}{9}$

2. $\frac{1}{9}$

3. $\frac{8}{9}$

4. $\frac{4}{9}$

When an object is shot from the bottom of a long, smooth inclined plane kept at an angle of 60$°$ with horizontal, it can travel a distance $x_1$ along the plane. But when the inclination is decreased to 30$°$ and the same object is shot with the same velocity, it can travel $x_2$ distance. Then $x_1$:$x_2$ will be:

1. $1:2\sqrt{3}$

2. $1:\sqrt{2}$

3. $\sqrt{2}:1$

4. $1:\sqrt{3}$

A force F= 20+10y acts on a particle in the y-direction where F is in Newton and y is in meter. Work done by this force to move the particle from y = 0 to y = 1m is:

1. 20 J

2. 30 J

3. 5 J

4. 25 J

A mass m is attached to a thin wire and whirled in a vertical circle. The wire is most likely to break when:

1. inclined at an angle of ${60}^0$ from vertical.

2. the mass is at the highest point.

3. the wire is horizontal.

4. the mass is at the lowest point.

A disc of radius 2 m and mass 100 kg rolls on a horizontal floor. Its centre of mass has speed of 20 cm/s. How much work is needed to stop it?

1. 1 J

2. 3 J

3. 30 J

4. 2 J

An object of mass 500 g initially at rest is acted upon by a variable force whose x-component varies with x in the manner shown. The velocities of the object at the points x = 8 m and x = 12 m would have the respective values of nearly:

1. 18 m/s and 24.4 m/s

2. 23 m/s and 24.4 m/s

3. 23 m/s and 20.5 m/s

4. 18 m/s and 20.5 m/s

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 AB = D. The height h is equal to :- 
      

1. $\frac{3}{2}D$

2. D

3. $\frac{7}{4}D$

4. $\frac{5}{4}D$