Two balls having different masses are moving towards each other with speed 30 m/s and 20 m/s as shown in figure (i). Their velocities after collision become 20 m/s and 30 m/s as shown in the figure (ii), then the coefficient of restitution is:

(1)  1

(2)  0.5

(3)  0.4

(4)  0.2

One sphere collides with another sphere of double its mass. While heavier mass was at rest initially. The ratio of their respective speeds after the collision will be (Coefficient of restitution = e)

1.  $\frac{1 - 2\mathrm{e}}{1 - \mathrm{e}}$

2.  $\frac{1 - 2\mathrm{e}}{1 + \mathrm{e}}$

3.  $\frac{1 - \mathrm{e}}{1 + 2\mathrm{e}}$

4.  $\frac{1 + 2\mathrm{e}}{1 - \mathrm{e}}$

A particle of mass ${\mathrm{m}}_1$ makes an elastic, one-dimensional collision with another stationary particle of mass ${\mathrm{m}}_2$. The fraction of kinetic energy of ${\mathrm{m}}_1$ carried away by ${\mathrm{m}}_2$ is

1.  $\frac{{\mathrm{m}}_1^2}{{\mathrm{m}}_2^2}$

2.  ${\left(\frac{{\mathrm{m}}_1^2}{{\mathrm{m}}_2^2}\right)}^{-1}$

3.  $\frac{{\mathrm{m}}_1 - {\mathrm{m}}_2}{{\left({\mathrm{m}}_1 + {\mathrm{m}}_2\right)}^2}$

4.  $\frac{4{\mathrm{m}}_1{\mathrm{m}}_2}{{\left({\mathrm{m}}_1 + {\mathrm{m}}_2\right)}^2}$

The potential energy \(\mathrm{U}\) of a system is given by $\mathrm{U}=$ $\mathrm{A}$ $-$ ${\mathrm{Bx}}^2$ (where \(\mathrm{x}\) is the position of its particle and \(\mathrm{A},\) \(\mathrm{B}\) are constants). The magnitude of the force acting on the particle is:
1. constant

2. proportional to \(\mathrm{x}\)

3. proportional to ${\mathrm{x}}^2$

4. proportional to $\left(\frac{1}{\mathrm{x}}\right)$

A person-1 stands on an elevator moving with an initial velocity of 'v' & upward acceleration 'a'. Another person-2 of the same mass m as person-1 is standing on the same elevator. The work done by the lift on the person-1 as observed by person-2 in time 't' is:

1.  $\mathrm{m}\left(\mathrm{g}\right)$ $+$ $\mathrm{a}\left(\mathrm{vt}\right)$ $+$ $\frac{1}{2}{\mathrm{at}}^2$

2.  $-\mathrm{mg}\left(\mathrm{vt}\right)$ $+$ $\frac{1}{2}{\mathrm{at}}^2$

3.  0

4.  $\mathrm{ma}\left(\mathrm{vt}\right)$ $+$ $\frac{1}{2}{\mathrm{at}}^2$

If a body of mass 2 kg is moved in the conservative field from point A to B in three different paths, then work done will be

(1)  ${\mathrm{W}}_{\mathrm{I}} < {\mathrm{W}}_{\mathrm{II}} < {\mathrm{W}}_{\mathrm{III}}$

(2)  ${\mathrm{W}}_{\mathrm{I}} > {\mathrm{W}}_{\mathrm{II}} > {\mathrm{W}}_{\mathrm{III}}$

(3)  ${\mathrm{W}}_{\mathrm{I}} = {\mathrm{W}}_{\mathrm{II}} = {\mathrm{W}}_{\mathrm{III}}$

(4)  ${\mathrm{W}}_{\mathrm{I}} > {\mathrm{W}}_{\mathrm{II}} = {\mathrm{W}}_{\mathrm{III}}$

Two particles moving initially in the same direction undergo a one-dimensional elastic collision. Their relative velocities before and after the collision are ${\overrightarrow{\mathrm{v}}}_1 \mathrm{and} {\overrightarrow{\mathrm{v}}}_2$. Which of the following is not correct?

1.  $\left|{\overrightarrow{\mathrm{v}}}_1\right| = \left|{\overrightarrow{v}}_2\right|$

2.  ${\overrightarrow{\mathrm{v}}}_1 = -{\overrightarrow{v}}_2$ only is the two are of equal mass

3.  ${\overrightarrow{\mathrm{v}}}_1.{\overrightarrow{v}}_2 = -{\left|{\overrightarrow{v}}_1\right|}^2$

4.  ${\overrightarrow{\mathrm{v}}}_1.{\overrightarrow{v}}_2 = -{\left|{\overrightarrow{v}}_2\right|}^2$

A particle is moving along the x-axis under a conservative force and its potential energy U varies with x co-ordinate as shown in the figure. Then force is positive at:

(1)  A

(2)  C, D 

(3)  B

(4)  D, E