The 'e' Equation

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.

A block of mass 'm' moving with speed 'v' collides with another block of mass '2m' which was at rest. The lighter block comes to rest after the collision. The coefficient of restitution is

1. $\frac{1}{2}$

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

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

4. $\frac{1}{5}$

Q. A $4\text{kg}$ ball moving to the right at $6\text{m/s}$ collides with a $5\text{kg}$ ball ahead of it and moving to the right at speed $2\text{m/s}$ . Find the magnitude of final velocity of the $4\text{kg}$ and $5\text{kg}$ balls respectively, if coefficient of restitution is $0.5$.

1. $4.4\text{m/s},3.4\text{m/s}$
2. $2.66\text{m/s},4.66\text{m/s}$
3. $3.33\text{m/s},5.33\text{m/s}$
4. $5\text{m/s},7\text{m/s}$

Q. A particle loses $25\%$ of its kinetic energy during head on collision with another identical particle initially at rest. The coefficient of restitution will be

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

Q. Two balls of mass $3\text{kg}$ and $6\text{kg}$ are moving towards each other with velocities $6\text{m/s}$ and $3\text{m/s}$ respectively on a frictionless surface. After colliding, the $3\text{kg}$ ball returns with velocity $3\text{m/s}$. Velocity of $6\text{kg}$ ball (after collision) and coefficient of restitution are respectively:

1. $1.5\text{m/s},0.5$
2. $2\text{m/s},0.20$
3. $3\text{m/s},1$
4. $2.2\text{m/s},0.3$

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. A ball falls from a height of $1\text{m}$ on the ground and jumps upto a height $H$. If the coefficient of restitution of the collision is $0.9$, then find the value of $H$.
(Take $g=10m/s^2$)

1. $0.9\text{m}$
2. $0.81\text{m}$
3. $1\text{m}$
4. $0.71\text{m}$

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 block of mass $m$ moving at a velocity $v$ collides head on with another block of mass $2m$ at rest. If the coefficient of restitution is $\frac{1}{2}$, find the velocities of the blocks after the collision.

1. $\frac{V}{2},\frac{V}{2}$

2. $\frac{V}{2},\frac{-V}{2}$

3. $\frac{V}{2},0$

4. $0,\frac{V}{2}$

Q. For the shown collision of the ball with a wall, the coefficient of restitution $e$ is found to be $\frac{1}{N}$. Then, the value of $N$ is

Q. A smooth sphere of mass $m$ is moving on a horizontal plane with velocity $3\hat{i}+\hat{j}$. It collides with a vertical wall which is parallel to vector$\hat{j}$. If the coefficient of restitution is $0.5$, the angle of velocity of the ball with repect to the normal to the wall after impact will be

1. ${30}^o$
2. ${\tan }^{-1}(\frac{1}{3})$
3. ${\cos }^{-1}\left(\frac{2}{\sqrt{13}}\right)$
4. ${\sin }^{-1}\left(\frac{2}{\sqrt{13}}\right)$

Q. A cubical block of side '$a$' moving on horizontal frictionless surface with velocity '$v$' hits a small nail [coefficient of restitution is e = 0] and rotates about it. Angular velocity of block after hitting the nail is

1. $\frac{v}{a}$
2. $\frac{3v}{4a}$
3. $\frac{2v}{3a}$
4. $\frac{v}{2a}$

Q. A block of mass $m$ moving at a velocity $v$ collides with another block of mass $2m$ at rest. The lighter block comes to rest after collision. Find the coefficient of restitution.

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

Q. A small ball moving with a velocity $10\text{m/s}$, horizontally (as shown in figure) strikes a rough horizontal surface having $\mu =0.5$. If the coefficient of restitution is $e=0.4$. Horizontal component of velocity of the ball after $1^{\text{st}}$ impact will be $(g=10{\text{m/s}}^2)$

1. $10\text{m/s}$
2. $8\text{m/s}$
3. $3\text{m/s}$
4. $4\text{m/s}$

Q. A body X with a momentum p collides with another identical stationary body Y one dimensionally. During the collision, Y gives an impulse J to body X. Then coefficient of restitution is

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

Q. A steel ball strikes a steel plate at an angle $\theta$ with the vertical. If the coefficient of restitution is $e$, the angle at which the rebound will take place is

1. $\theta$
2. ${\tan }^{-1}\left[\frac{\tan \theta }{e}\right]$
3. $e\tan \theta$
4. ${\tan }^{-1}\left[\frac{e}{\tan \theta }\right]$

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 block of mass $m$ moving at a velocity $v$ collides head on with another block of mass $2m$ at rest. If the coefficient of restitution is $\frac{1}{2}$, find the velocities of the blocks after the collision.

1. $\frac{V}{2},\frac{V}{2}$

2. $\frac{V}{2},\frac{-V}{2}$

3. $\frac{V}{2},0$

4. $0,\frac{V}{2}$

Q.

A ball collides directly with another ball at rest and is itself brought to rest by the impact. If one fourth of initial kinetic energy is destroyed in the collision, the coefficient of restitution of the collision is

1. $\frac{1}{2}$

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

3. $\frac{3}{4}$

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

Q. Two equal spheres $A$ and $B$ lie on a smooth horizontal circular groove at opposite ends of a diameter. At time $t=0$, $B$ is projected along the groove and it first impinges on $A$ at time $t=T_1$ and again at time $t=T_2$. If $e$ is the coefficient of restitution, the ratio $\frac{T_2}{T_1}$ is

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

Q. A block $A$ of mass $2\text{kg}$ moving with speed of $15\text{m/s}$, collides with a stationary block $B$ of mass $5\text{kg}$. If the coefficient of restitution is $0.6$, find the magnitude of velocity of blocks $A$ and $B$ after the collision respectively.

1. $5.2\text{m/s}$ and $10.5\text{m/s}$
2. $3.5\text{m/s}$ and $6.85\text{m/s}$
3. $10.5\text{m/s}$ and $5.2\text{m/s}$
4. $2.15\text{m/s}$ and $6.85\text{m/s}$

Q. A teacher gave two statements about coefficient of restitution,
Statement 1: For a perfectly elastic collision $e=1$.
Statement 2: Coefficient of restitution is the ratio of relative velocity of body 2 w.r.t body 1 after collision to relative velocity of body 1 w.r.t body 2 before collision.

Which of the following option is correct?

1. Statement 1 and statement 2 both are true.
2. Only statement 1 is true.
3. Only statement 2 is true.
4. Statement 1 is true and data is insufficient for statement 2.

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

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.

In the previous question, if the collision is inelastic with, $e=\frac{2}{5}$, then repeat the above questions.

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 loss in $KE$ due to collision.

1. $2.22\text{J}$
2. $15.12\text{J}$
3. $4.52\text{J}$
4. $16.74\text{J}$

Q. A $4\text{kg}$ ball moving to the right at $6\text{m/s}$ collides with a $5\text{kg}$ ball ahead of it and moving to the right at speed $2\text{m/s}$ . Find the magnitude of final velocity of the $4\text{kg}$ and $5\text{kg}$ balls respectively, if coefficient of restitution is $0.5$.

1. $4.4\text{m/s},3.4\text{m/s}$
2. $2.66\text{m/s},4.66\text{m/s}$
3. $3.33\text{m/s},5.33\text{m/s}$
4. $5\text{m/s},7\text{m/s}$