The gravitational field in a region is given by $\overrightarrow{\mathrm{g}} = 4\hat{\mathrm{i}} + 4\hat{\mathrm{j}}$. Work done by this field is zero, when a particle is moved along the line

(1) 3x + 4y = 2

(2) 3x + 3y = 5

(3) x + 3y = 2

(4) 3y + x = 1

Five balls are placed one after another along a straight line as shown in the figure. Initially, all the balls are at rest. Then the second ball is projected with speed ${\mathrm{v}}_0$ towards the third ball. Mark the correct statement(s). (Assume all collisions to be head-on and elastic):
     

1. The total number of collisions in the process is 5.

2. The velocity of separation between the first and fifth ball after the last possible collision is ${\mathrm{v}}_{0.}$

3. Finally, three balls remain stationary.

4. All of the above are correct.

A body of mass 2 kg moving with a velocity of 3 m/s collides with a body of mass of 1 kg moving with a velocity of 4 m/s in the opposite direction. If the collision is head-on and completely inelastic, then the wrong statement is:

1. Both bodies move together with a velocity (2/3) m/s.

2. The momentum of the system is 2 kg-m/s throughout.

3. The momentum of the system is 10 kg-m/s.

4. The loss of KE for the system is (49/3) J.

A smooth sphere of mass M, moving with velocity u, directly collides elastically with another sphere of mass m at rest. After the collision, their final velocities are V and v, respectively. The value of v is:

1. \(\frac{2 um}{m}\)

2. \(\frac{2 um}{M}\)

3. \(\frac{2 u}{1 + {m \over M}}\)

4. \(\frac{2 u}{1 + {M \over m}}\)

A block of mass m is moving with speed v towards a spring block system. If the collision is perfectly inelastic, then the maximum compression in the spring will be:

1.  $\mathrm{v}\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$

2.  $\mathrm{v}\sqrt{\frac{2\mathrm{m}}{\mathrm{k}}}$

3.  $\mathrm{m}\sqrt{\frac{\mathrm{v}}{2\mathrm{k}}}$

4.  $\mathrm{v}\sqrt{\frac{\mathrm{m}}{2\mathrm{k}}}$

Two identical balls of mass 1 kg each moving towards each other with velocities of 3 m/s and 5 m/s respectively. The magnitude of the velocities of respective balls after the perfectly elastic head-on collision is

(1) 3 m/s, 5 m/s

(2) 5 m/s, 3 m/s

(3) 4 m/s, 4 m/s

(4) 6 m/s, 2 m/sIf the two masses are equal

A body is under uniform circular motion, its kinetic energy.

1. depends on the direction of velocity.
2. depends on acceleration.
3.  remains unchanged.
4.  all of these.

A particle of mass m is moving in a circular path of constant radius r such that its centripetal acceleration ${\mathrm{a}}_{\mathrm{r}}$ is varying with time t as ${\mathrm{krt}}^2$,where k is constant. The power delivered to the particle by the force acting on it is

(1)  $2{\mathrm{\pi kmr}}^2$

(2)  ${\mathrm{kmr}}^2\mathrm{t}$

(3)  ${\mathrm{kmr}}^2{\mathrm{t}}^5$

(4)  Zero

The ratio of the kinetic energy of a particle projected from the ground at the  highest point to point of projection is 1/4. The angle of projection with horizontal is

(1)  30°

(2)  60°

(3)  45°

(4)  53°

A bob is suspended from a peg on a wall by a massless string of length L. If it is given velocity $\sqrt{2gL}$ at the lowest point, then angular amplitude with which body will oscillate is

(1)  60°

(2)  90°

(3)  75°

(4)  30°