Velocity of Separation and Approach

Q. A prticle of mass $m$ moves with velocity $v_0=20\text{m/s}$ towards a large wall that is moving with velocity $v=5\text{m/s}$ towards the particle as shown. If the particle collides with the wall elastically, then find the speed of the particle just after collision. (Assume collision with the wall is elastic)

1. $25\text{m/s}$
2. $30\text{m/s}$
3. $22\text{m/s}$
4. $20\text{m/s}$

Q. A ball of mass $m$ moving with speed $u$ undergoes a head on perfect elastic collision with mass $nm$ initially at rest. The fraction of the energy transferred to the second ball by the first ball is

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

Q. A ball moving at a speed of $3\text{m/s}$ approaches a wall moving towards it with a speed of $3\text{m/s}$. After the perfectly elastic collision, the speed of the ball will be

1. $3\text{m/s}$
2. $6\text{m/s}$
3. $9\text{m/s}$
4. $0$

Q. A particle of mass $m$ is dropped from a height $h$ above the ground. At the same time another particle of the same mass is thrown vertically upwards from the ground with a speed of $\sqrt{2gh}$ . If they collide head-on completely inelastically, the time taken for the combined mass to reach the ground, in units of $\sqrt{\frac{h}{g}}$ is

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

Q. Body $A$ of mass $m$ and body $B$ of mass $3m$ move towards each other with velocities $V$ and $2V$ respectively from the positions as shown, along a smooth horizontal circular track of radius $r$. After the first elastic collision, they will collide again after the time

1. $\frac{\pi r}{2V}$
2. $\frac{\pi r}{V}$
3. $\frac{2\pi r}{V}$
4. $\frac{2\pi r}{3V}$

Q. Two equal mass objects are moving in a circular path with constant speed as shown in figure. After how many collisions they will again meet at $A$.

1. $1$
2. $2$
3. $4$
4. $3$

Q. A body of mass $m_1$ moving at a constant speed undergoes an elastic collision with a body of mass $m_2$ initially at rest. The ratio of the kinetic energy of mass $m_1$ after the collision to that before the collision is

1. ${\left(\frac{m_1+m_2}{m_1-m_2}\right)}^2$
2. ${\left(\frac{2m_1}{m_1+m_2}\right)}^2$
3. ${\left(\frac{2m_2}{m_1+m_2}\right)}^2$
4. ${\left(\frac{m_1-m_2}{m_1+m_2}\right)}^2$

Q. 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. $\frac{2uM}{m}$
2. $\frac{2um}{M}$
3. $\frac{2u}{1+\frac{m}{M}}$
4. $\frac{2u}{1+\frac{M}{m}}$

Q. A ball is dropped from a height $h$ on a floor, where the coefficient of restitution is $e$. Find the time interval between the instant of drop and second collision with ground.

1. $\sqrt{\frac{2h}{g}}(1+e)$
2. $\sqrt{\frac{2h}{g}}(1+\frac{e}{2})$
3. $\sqrt{\frac{2h}{g}}(1+2e)$
4. None of these

Q. Calculate the escape velocity from the moon. The mass of the moon is $7.4\times {10}^{22}\text{kg}$ and the radius of the moon is $1740\text{km}$

1. $1.4\text{km/hr}$
2. $2.4\text{km/s}$
3. $1.4\text{km/s}$
4. $2.4\text{km/hr}$

Q. Balls A and B are thrown from two points lying on the same horizontal plane separated by a distance 120 m. Which of the following statements (s) is/are correct?

1. The two balls can never meet
2. The balls can meet if the ball B is thrown 1 s later
3. The two balls meet at a height of 45 m
4. None of the above

Q. A small ball of mass $m$ is connected by an inextensible massless string of length $l$ with another ball of mass $M=4\text{m}$. They are released with zero tension in the string from a height $h$ as shown in the figure. The time after which string becomes taut for the first time after the release(after the mass $M$ collides with the ground ) is $\frac{l}{\sqrt{ngh}},$ where $n=$
(Assume all collisions to be elastic)

Q. A ball is projected upwards from the top of a tower with a velocity $50\text{m/s}$ making an angle ${30}^{\circ }$ with the horizontal. The height of the tower is $70\text{meters}$. After how much time from the instant of throwing, the ball will reach the ground?

1. $2\text{sec}$
2. $5\text{sec}$
3. $7\text{sec}$
4. $9\text{sec}$

Q. A small ball of mass $m$ is connected by an inextensible massless string of length $l$ with another ball of mass $M=4m$. They are released with zero tension in the string from a height $h$ as shown in the figure. The time $t$ after which the string becomes taut for the first time after the mass $m$ collides with the ground is (Assume all the collisions to be elastic)

1. $t=\frac{3l}{\sqrt{2gh}}$
2. $t=\frac{l}{\sqrt{2gh}}$
3. $t=\frac{l}{\sqrt{3gh}}$
4. $t=\frac{l}{\sqrt{gh}}$

Q. A particle is projected with a speed $u$ at an angle $\theta$ with horizontal from point $A$. It strikes elastically with a vertical wall at height $h/2$. It rebounds and reaches maximum height $h$ and falls back on the ground at point $B$ as shown in Figure. Distances from $A$ to wall and from wall to $B$ are $x_1$ and $x_2$, and time to cover is $t_1$ and $t_2$ respectively. Match the values in column I with the expressions in column II.


$$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{i.}\sqrt{2}& \text{a.}\frac{x_2-x_1}{x_2+x_1}\text{or}\frac{x_2+x_1}{x_2-x_1}\\ \text{ii.}\frac{1}{\sqrt{2}}& \text{b.}\frac{t_2-t_1}{t_2+t_1}\text{or}\frac{t_2+t_1}{t_2-t_1}\\ \text{iii.}1& \text{c.}\frac{u\sin \theta }{g(t_2+t_1)}\\ \text{iv.}\frac{1}{2}& \text{d.}\frac{u\cos \theta (t_1+t_2)}{x_1+x_2}\end{array}$$

1. $\text{(i) a, b (ii) a, b (iii) d (iv) c}$
2. $\text{(i) a (ii) b (iii) d (iv) c}$
3. $\text{(i) a, b (ii) a, b (iii) d (iv) c}$
4. $\text{(i) a (ii) a, c (iii) d (iv) b}$

Q.

Q. The velocity of a projectile at the initial point A is $(2\hat{i}+3\hat{j})m/s$. Its velocity (in $m/s$) at point B is

1. $-2\hat{i}-3\hat{j}$
2. $-2\hat{i}+3\hat{j}$
3. $2\hat{i}-3\hat{j}$
4. $2\hat{i}+3\hat{j}$

Q. Two equal mass objects are moving in a circular path with constant speed as shown in the figure. The collision is elastic everytime. They meet at $A$ again on $n^{th}$ collision, find the value of $n?$

[Assume, collisions are elastic in nature]

1. $1$
2. $2$
3. $3$
4. $4$

Q. A particle of mass 100g moving at an initial spped u collides with another particle of same mass kept initially at rest. If the total kinetic energy becomes 0.5J after the collision, what could be the minimum and the maximum value of u.

1. 2 m/s, 2$\sqrt{2}$m/s
2. $2\sqrt{2}$m/s , 2 m/s
3. 2 m/s, 4 m/s
4. 2 m/s , 2 m/s

Q.

A large body of mass 100 kg moving with a velocity of 10.0 m/s collides elastically with a small body of mass 100 g at rest. The velocity of the small body is

1. 20 m/s

2. 5 m/s

3. 10 m/s

4. 25 m/s

Q.
In the diagram shown block $\text{a}$ moves to right and $\text{f}$ and $\text{g}$ move towards left with a speed $v$. All collisions are elastic, and the blocks are identical. Then after sufficient time

1. $\text{a}$ will move to left with speed $v$
2. $a$ will be at rest
3. $\text{f}$ will keep moving toward left
4. $\text{g}$ will be at rest

Q. A particle of mass $m_1$ and moving with velocity $v_1$ collides head on with another stationary particle of mass $m_2$elastically. If after the collision their velocities are $v_1$ and $v_2$ then under the condition $m_1=m_2$ their values will be :

1. $v_1=0,v_2=0$
2. $v_1=u_1,v_2=u_2$
3. $v_1=0,v_2=u_1$
4. $v_2=0,v_1=u_1$

Q. A ball moving at a speed of $3\text{m/s}$ approaches a wall moving towards it with a speed of $3\text{m/s}$. After the perfectly elastic collision, the speed of the ball will be

1. $3\text{m/s}$
2. $6\text{m/s}$
3. $9\text{m/s}$
4. $0$

Q. If a ball of mass $0.4kg$ moving with a velocity of $3m{s}^{-1}$ collides elastically with another ball of mass $0.6kg$ which is at rest, Their velocities after collision will be

1. $V_1=0.3m{s}^{-1},V_2=1.4m{s}^{-1}$
2. $V_1=2.4m{s}^{-1},V_2=0.6m{s}^{-1}$
3. $V_1=1.4m{s}^{-1},V_2=0.3m{s}^{-1}$
4. $V_1=1.2m{s}^{-1},V_2=1.2m{s}^{-1}$

Q. Assertion :The relative velocity of the two particles in head-on elastic collision is unchanged both in magnitude and direction. Reason: The relative velocity is unchanged in magnitude but gets reversed in direction.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q. Two particles A and B, move with constant velocities $\overrightarrow{v_1}$ and $\overrightarrow{v_2}$. At the initial moment their position vectors are $\overrightarrow{r_1}$ and $\overrightarrow{r_2}$ respectively. The condition for particles A and B for their collision is:

1. $\overrightarrow{r_1}-\overrightarrow{r_2}=\overrightarrow{v_1}-\overrightarrow{v_2}$
2. $\frac{\overrightarrow{r_1}-\overrightarrow{r_2}}{|\overrightarrow{r_1}-\overrightarrow{r_2}|}=\frac{\overrightarrow{v_2}-\overrightarrow{v_1}}{|\overrightarrow{v_2}-\overrightarrow{v_1}|}$
3. $\overrightarrow{r_1}\times \overrightarrow{v_1}=\overrightarrow{r_2}\times \overrightarrow{v_2}$
4. $\overrightarrow{r_1}\cdot \overrightarrow{v_1}=\overrightarrow{r_2}\cdot \overrightarrow{v_2}$

Q. A solid sphere is rolling purely on a rough horizontal surface with speed of center $u=12$ m/s. It collides inelastically with a smooth vertical wall at a certain moment, the coefficient of restitution being $\frac{1}{2}$. How long (in sec) after the collision, the sphere will begin pure rolling?
[coefficient of friction between the sphere and the ground is $\frac{3}{35}$]

1. 16
2. 12
3. 6
4. 13

Q. A particle of mass $2m$ is projected at an angle of ${45}^o$ with horizontal with a velocity of $20\sqrt{2}m/s$. After $1s$ explosion takes place and the particle is broken into two equal pieces. As a result of explosion one part comes to rest. Find the maximum height attained by the other part. Take $g=10m/s^2$:

1. $25m$
2. $50m$
3. $15m$
4. $35m$

Q. A particle is projected from the ground at $t=0$ so that on its way it just clears two vertical walls of equal height on the ground. The particle was projected with initial velocity $u$ and at angle $\theta$ with the horizontal. If the particle passes just grazing top of the wall at time $t=t_1$ and $t=t_2$ then calculate.
a. the height of the wall.
b. the time $t_1$ and $t_2$ in terms of height of the wall. Write the expression for calculating the range of this projecting and separation between the walls.