A ball is thrown vertically downwards from a height of 20 m with an initial velocity v₀. It collides with the ground, loses 50% of its energy in a collision, and rebounds to the same height. The initial velocity v₀ is: 
[Take, g = 10 ms^-1]

1. 14 ms^-1

2. 20 ms^-1

3. 28 ms^-1

4. 10 ms^-1

Two similar springs P and Q have spring constants k_P and k_Q, such that k_P> k_Q. They are stretched, first by the same amount (case a), then by the same force (case b). The work done by the springs W_P and W_Q are related as, in case (a) and case (b), respectively :
1. \(W_P=W_Q;W_P>W_Q\)
2. \(W_P=W_Q;W_P=W_Q\)
3. \(W_P>W_Q;W_P<W_Q\)
4. \(W_P<W_Q;W_P<W_Q\)

A block of mass 10 kg, moving in the x-direction with a constant speed of 10 ms^-1 is subjected to a retarding force F = 0.1x J/m during its travel from x = 20 m to 30 m. Its final K.E. will be:

A particle of mass m is driven by a machine that delivers a constant power of k watts. If the particle starts from rest, the force on the particle at time t is:

1. $\sqrt{\frac{mk}{2}}{t}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

2. $\sqrt{mk}{t}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

3. $\sqrt{2mk}{t}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

4. $\frac{1}{2}$$\sqrt{mk}{t}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Two particles of masses m₁ and m₂ move with initial velocities u₁ and u₂ respectively. On collision, one of the particles gets excited to a higher level, after absorbing energy E. If the final velocities of particles are v₁ and v₂, then we must have:

1. ${\mathrm{m}}_1^2{\mathrm{u}}_1+{\mathrm{m}}_2^2{\mathrm{u}}_2-\mathrm{E}={\mathrm{m}}_1^2{\mathrm{v}}_1+{\mathrm{m}}_2^2{\mathrm{v}}_2$

2. $\frac{1}{2}{\mathrm{m}}_1{\mathrm{u}}_1^2+\frac{1}{2}{\mathrm{m}}_2{\mathrm{u}}_2^2=\frac{1}{2}{\mathrm{m}}_1{\mathrm{v}}_1^2+\frac{1}{2}{\mathrm{m}}_2{\mathrm{v}}_2^2$

3. $\frac{1}{2}{\mathrm{m}}_1{\mathrm{u}}_1^2+\frac{1}{2}{\mathrm{m}}_2{\mathrm{u}}_2^2-\mathrm{E}=\frac{1}{2}{\mathrm{m}}_1{\mathrm{v}}_1^2+\frac{1}{2}{\mathrm{m}}_2{\mathrm{v}}_2^2$

4. $\frac{1}{2}{\mathrm{m}}_1^2{\mathrm{u}}_1^2+\frac{1}{2}{\mathrm{m}}_2^2{\mathrm{u}}_2^2+\mathrm{E}=\frac{1}{2}{\mathrm{m}}_1^2{\mathrm{v}}_1^2+\frac{1}{2}{\mathrm{m}}_2^2{\mathrm{v}}_2^2$

On a frictionless surface, a block of mass M moving at speed v collides elastically with another block of the same mass M which is initially at rest. After the collision, the first block moves at an angle $\theta$ to its initial direction and has a speed $\frac{v}{3}$. The second block’s speed after the collision will be:

1. $\frac{2\sqrt{2}}{3}v$

2. $\frac{3}{4}v$

3. $\frac{3}{\sqrt{2}}v$

4. $\frac{\sqrt{3}}{2}v$

A uniform force of \((3 \hat{i} + \hat{j})\) newton acts on a particle of mass 2 kg. Hence the particle is displaced from position \((2 \hat{i} + \hat{k})\) meter to position \((4 \hat{i} + 3 \hat{j} - \hat{k})\) meter. The work done by the force on the particle is:

1. 6 J
2. 13 J
3. 15 J
4. 9 J

The potential energy of a particle in a force field is U=$\frac{A}{r^2}-\frac{B}{r}$ where A and B are positive constants and r is the distance of the particle from the center of the field. For stable equilibrium, the distance of the particle is:

1. B/A

2. B/2A

3. 2A/B

4. A/B

Two spheres A and B of masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$, respectively, collide. A is at rest initially and B is moving with velocity v along the x-axis. After collision, B has a velocity $\frac{\mathrm{v}}{2}$ in a direction perpendicular to the original direction. The mass A moves after collision in the direction:
 

1. same as that of B.

2. opposite to that of B.

3.  $\mathrm{\theta }={\tan }^{-1}\left(\frac{1}{2}\right)$ to the positive x-axis.

4. $\mathrm{\theta }={\tan }^{-1}\left(\frac{-1}{2}\right)$ to the positive x-axis

A car of mass m starts from rest and accelerates so that the instantaneous power delivered to the car has a
constant magnitude $P_0$. The instantaneous velocity of this car is proportional to -

1.  $t^{1/2}$

2.  $t^{-1/2}$

3.  $t/\sqrt{m}$

4.  $t^2{P}_0$