Gravitational Potential of a Ring

Q. Two balls of masses $3\text{kg}$ and $5\text{kg}$ are moving towards each other with speeds $5\text{m/s}$ and $3\text{m/s}$ respectively on a frictionless surface as shown in figure. After collision, the $5\text{kg}$ ball retraces its path with speed $2\text{m/s}$. Find the coefficient of restitution.

1. $1$
2. $\frac{1}{3}$
3. $\frac{2}{3}$
4. $\frac{8}{3}$

Q. Two balls of masses $2\text{kg}$ and $5\text{kg}$ moving on a frictionless surface, collide head-on with speeds $2\text{m/s}$ and $3\text{m/s}$ as shown in figure. If the coefficient of restitution is $0.25$, then the kinetic energy lost by the balls is

1. $0.65\text{J}$
2. $0.35\text{J}$
3. $0\text{J}$
4. $16.74\text{J}$

Q. A ball moving with velocity $2{\text{ms}}^{-1}$ collides head-on with another stationary ball of double the mass. If the coefficient of restitution is $0.5$, then their velocities (in ${\text{ms}}^{-1}$) after collision will be

1. $0,1$
2. $1,1$
3. $1,0.5$
4. $0,2$

Q. A particle of mass $3\text{kg}$ moving with a velocity of $6\hat{i}\text{m/s}$ collides head on with another particle of mass $5\text{kg}$ moving with a velocity of $-2\hat{i}\text{m/s}$. If after the collision, first particle has a speed of $1\text{m/s}$ along the negative $X$ axis, find the coefficient of restitution.

1. $0.4$
2. $0.5$
3. $0.6$
4. $0.3$

Q. Two billiard balls undergo a head-on collision. Ball $1$ is twice as heavy as ball $2$. Initially, ball $1$ moves with a speed $u$ towards ball $2$ which is at rest. Immediately after the collision, ball $1$ travels at a speed of $\frac{u}{3}$ in the same direction. What type of collision has occurred?

1. Inelastic
2. Elastic
3. Completely inelastic
4. cannot be determined from the information

Q. A $3\text{kg}$ ball moving in rightward direction on a horizontal surface with speed $4\text{m/s}$ collides with a $4\text{kg}$ ball moving ahead of it, in rightward direction with a speed of $2\text{m/s}$. If the coefficient of restitution is $0.6$, find the final speed of $3\text{kg}$ and $4\text{kg}$ balls respectively in $\text{m/s}$.

1. $6.67\text{m/s}$ and $8.27\text{m/s}$
2. $1.17\text{m/s}$ and $2.27\text{m/s}$
3. $4\text{m/s}$ and $5\text{m/s}$
4. $2.17\text{m/s}$ and $3.37\text{m/s}$

Q. A sphere of mass $m$ moving with a constant velocity $u$ hits another stationary sphere of some mass. If $e$ is the coefficient of restitution, then ratio of velocities of the two spheres $\frac{v_1}{v_2}$ after collision will be

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

Q. Particle 'A' moves with speed $10\text{m/s}$ in a frictionless circular fixed horiozontal pipe of radius $5$ m and strikes with ${}^{\prime }{B}^{\prime }$ of double mass that of A. Coefficient of resitution is $\frac{1}{2}$ and particle 'A' starts its journey at $t=0$. The time at which second collision occurs is:

1. $\frac{\pi }{2}s$
2. $\frac{2\pi }{3}s$
3. $\frac{5\pi }{2}s$
4. $4\pi s$

Q. A body of mass $4\text{kg}$ moving with a speed of $6\text{m/s}$ collides with another body of mass $8\text{kg}$ at rest. The lighter body comes to rest immediately after the collision. Find the coefficient of restitution $\left(e\right)$ for the collision.

1. $0.25$
2. $0.6$
3. $0.3$
4. $0.5$

Q.

A ball strikes a wall with a velocity $\overrightarrow{u}$ at an angle $\theta$ with the normal to the wall surface and rebounds from it at an angle $\beta$ with the surface. Then:

1. If wall is smooth, $(\theta +\beta )$ > ${90}^{\circ }$

2. If the wall is smooth, coefficient of restitution $=\frac{tan\beta }{cot\theta }$

3. If wall is smooth, coefficient of restitution < $\frac{tan\beta }{cot\theta }$

4. None of these