Bouncing Ball Example

Q. A ball after freely falling from a height of $4.9\text{m}$ strikes a fixed horizontal plane. If the coefficient of restitution is $\frac{3}{4}$, the ball will strike second time with the plane after

1. $0.5\text{s}$
2. $1\text{s}$
3. $1.5\text{s}$
4. $2\text{s}$

Q. A ball is dropped from a height $h$ on to a floor. If in each collision its speed becomes $e$ times of its striking value, then time taken by ball to stop rebounding is (Here, $e$ is coefficient of restitution between the ball and the floor)

1. $\sqrt{\frac{2h}{g}}\left(\frac{1+e}{1-e}\right)$
2. $\sqrt{\frac{h}{g}}\left(\frac{2-e}{2+e}\right)$
3. $\sqrt{2gh}\left(\frac{2-e}{2+e}\right)$
4. $\sqrt{\frac{h}{g}}\left(\frac{e}{1-e}\right)$

Q. A body falls from a height 'h' on a horizontal surface and rebounds. Then it falls again and again rebounds and so on. If the restitution coefficient is $\frac{1}{3}$, the total time taken by the body to come to rest is

1. 
2. 
3. 
4. 

Q. A ball initially at rest, is dropped from building of height $10\text{m}$ to ground as shown in the figure below. If the coefficient of restitution is $e=0.7$ and after collision, the ball rebounds back up to height $h$, find the value of $h$. (Take $g=9.8{\text{m/s}}^2)$

1. $10\text{m}$
2. $4.9\text{m}$
3. $9.8\text{m}$
4. $\text{None of these}$

Q. A ball is projected with an initial speed $u$ at an angle $\theta$ from the horizontal ground. The coefficient of restitution between the ball and the ground is $e$. Find the position from the starting point when the ball will land on the ground for the $2^{nd}$ time.

1. $\frac{(1-e^2)u^2\sin 2\theta }{g}$
2. $\frac{e^2{u}^2\sin 2\theta }{g}$
3. $\frac{(1-e)u^2\sin \theta \cos \theta }{g}$
4. $\frac{(1+e)u^2\sin 2\theta }{g}$

Q.

A steel ball falls from a height ‘h’ on a floor for which the coefficient of restitution is e. The height attained by the ball after the first rebounds is

1. $h{e}^2$
2. $h/e$
3. $h/e^2$
4. $he$

Q. A body falls from a height 'h' on a horizontal surface and rebounds. Then it falls again and again rebounds and so on. If the restitution coefficient is $\frac{1}{3}$, the total time taken by the body to come to rest is

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

Q. A ball falls on the ground from a height of $2.0\text{m}$ and rebounds up to a height of $1.5\text{m}$. Find the coefficient of restitution for the collision of the ball with the ground. Take $g=10{\text{m/s}}^2$.

1. $1$
2. $0$
3. $\frac{1}{2}$
4. $\frac{\sqrt{3}}{2}$

Q. A ball falling freely from a height of $4.9\text{m}$, hits a horizontal surface. If $e=\frac{3}{4}$, then the ball will hit the surface, second time after

1. $1.5\text{s}$
2. $2.0\text{s}$
3. $3.0\text{s}$
4. $1.0\text{s}$

Q.

A 2g block attached to an ideal spring with a spring constant of 80N/m oscillates on a horizontal frictionless surface.When the spring is 4.0 cm shorter than its equilibrium length, the speed of 17^1/2m/s.The greatest speed of the block is

1)4m/s

2)6m/s

3)5m/s

4)9m/s

5)3m/s

Q. A ball is projected from a point in one of the two smooth parallel vertical walls against the other in a plane perpendicular to both. After being reflected at each wall impinge again on the second at a point in the same plane as it started. The distance between two walls is $a.b$ is the free range and $e$ be the coefficient of restitution:

1. The total time taken in moving from $O$ to $C$ is $\frac{a}{e^{2}u}(e^2+e+1)$
2. The free range on the horizontal plane $b=\frac{2uv}{g}$
3. $b{e}^2=a(e^2+e+1)$
4. All of these

Q.

What is the percentage change in the momentum of the body, if the mass of a body is doubled and its velocity is reduced by half?

1. 0 %

2. 10 %

3. 50 %

4. 100 %

Q. A particle is projected from a point A that is at a distance of $4R$ from the centre of the earth with speed $V_1$ in a direction making ${30}^{\circ }$ with the line joining the centre of the earth and point A as shown in figure. Consider the gravitational interaction only between these two bodies (use $\frac{GM}{R}=6.4\times {10}^7{m}^2/s^2$). The speed $V_1$ if particle passes grazing the surface of the earth is:

1. $2\sqrt{2}\times {10}^{3}m/s$
2. $4\sqrt{2}\times {10}^{3}m/s$
3. $4\times {10}^{3}m/s$
4. $4\sqrt{3}\times {10}^{3}m/s$

Q. A ball is projected in a direction inclined to the horizontal and it bounces on a horizontal plane. If the range of first rebound is $R$ and coefficient of restitution is $e$, then range of the next rebound is

1. $R^{\prime }=eR$
2. $R^{\prime }=e^{2}R$
3. $R^{\prime }=R$
4. $R^{\prime }=\frac{R}{e}$

Q.

A ball is dropped from a height of $20m$. If the coefficient of restitution is $0.9$, what will be the height attained after the first bounce?

1. $1.62m$

2. $16.2m$

3. $18m$

4. $14m$

Q.

A ball dropped freely takes 0.2 seconds to cross the last 6 m distance before hitting the ground. total time of fall is ? ( g =10m/s^{2 )}

^2.9s

^3.1s

^2.7s

^0.2s

Q. A small ball rolls off the top landing of the staircase. If strikes the mid- point of the first step and then the mid - point of the second step. The steps are smooth, and identical in height and width. The coefficient of restitution between the ball and the first step is

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

Q. A ball is dropped from a height $h$ on to a floor. If in each collision its speed becomes $e$ times of its striking value, then total change in momentum of ball when it stops rebounding is (Here, $e$ is coefficient of restitution between the ball and the floor)

1. $m\sqrt{2gh}\left(\frac{e}{e(1+e)}\right)$
2. $\sqrt{2mgh}\left(\frac{e}{1-e}\right)$
3. $m\sqrt{2gh}\left(\frac{2e}{1+e}\right)$
4. $m\sqrt{2gh}\left(\frac{1+e}{1-e}\right)$

Q. Three concentric spherical shells have radii a, b, and c(a < b < c) and have surface charge densities $\sigma$, $-\sigma$ and$\sigma$ respectively. If $V_A$, $V_B$ and $V_C$ denote the potentials of the three shells, then, for c = a + b, we have :

1. $V_C$ = $V_B$ = $V_A$
2. $V_C$ = $V_A\ne V_B$
3. $V_C$ = $V_B\ne V_A$
4. $V_C\ne V_B\ne V_A$

Q.

A body falls from a height h on a horizontal surface and rebounds. Then it falls again and rebounds and so on. If the restitution coefficient is $\frac{1}{3}$, the total distance covered by the body before it comes to rest is

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

2. $3h$

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

4. $2h$

Q. A ball is thrown upwards from the bottom of a fixed chamber of height $20\text{m}$ with a speed of $10\text{m/s}$. The interior of the chamber has no gravity. The ball strikes the ceiling of the chamber and rebounds. After $6\text{seconds}$, gravity is reintroduced in the chamber. The movement of the ball gradually slows down over time and the ball comes to rest. Which of the following is the correct displacement vs time graph for the movement of the ball inside the chamber?

1. 
2. 
3. 
4. 

Q.

A ball is dropped from a hieght of 5m onto a Sandy floor and penetrates the sand upto 10 cm before coming to rest . Find the retardation of the ball in sand assuming it to be uniform.

Q. A body falls from a height 'h' on a horizontal surface and rebounds. Then it falls again and again rebounds and so on. If the restitution coefficient is $\frac{1}{3}$, the total distance covered by the body before it comes to rest is

1. 
2. 
3. 
4. 

Q. A ball is thrown upwards from the bottom of a fixed chamber of height $20\text{m}$ with a speed of $10\text{m/s}$. The interior of the chamber has no gravity. The ball strikes the ceiling of the chamber and rebounds. After $6\text{seconds}$, gravity is reintroduced in the chamber. The movement of the ball gradually slows down over time and the ball comes to rest. Which of the following is the correct displacement vs time graph for the movement of the ball inside the chamber?

1. 
2. 
3. 
4. 

Q.

A ball is dropped from a height. If it takes 0.2s to cross the last 6 m before hitting the ground, find the height from which it was dropped.

Q. An object is placed $30cm$ away from a convex lens of focal length $10cm$ and a sharp image is formed on a screen. Now a concave lens is placed in contact with the convex lens. The screen now has to be moved by $45cm$ to get a sharp image again. The magnitude of focal length of the concave lens is (in $cm$):

1. 72
2. 60
3. 36
4. 20

Q. A ball initially at rest falls from a height $h=2.5\text{m}$. After collision with surface having value of coefficient of restitution $e=0.6$, it rebounds back. Find the rebound velocity of ball.

1. $2\text{m/s}$
2. $4.2\text{m/s}$
3. $6\text{m/s}$
4. $11.66\text{m/s}$

Q. In the experimental arrangement shown in figure, the area of cross-section of the wide and narrow portions of the tube are $5{\text{cm}}^2$ and $2{\text{cm}}^2$ respectively. The rate of flow of water through the tube is $500{\text{cm}}^3{\text{s}}^{-1}$. The difference of mercury levels in the U-tube is:

1. $0.97\text{cm}$
2. $1.96\text{cm}$
3. $0.67\text{cm}$
4. $4.67\text{cm}$

Q. A ball is dropped from a height $h$ on to a floor. If in each collision its speed becomes $e$ times of its striking value, then time taken by ball to stop rebounding is (Here, $e$ is coefficient of restitution between the ball and the floor)

1. $\sqrt{\frac{2h}{g}}\left(\frac{1+e}{1-e}\right)$
2. $\sqrt{\frac{h}{g}}\left(\frac{2-e}{2+e}\right)$
3. $\sqrt{2gh}\left(\frac{2-e}{2+e}\right)$
4. $\sqrt{\frac{h}{g}}\left(\frac{e}{1-e}\right)$

Q. Two similar balls of mass $m$ are hung from silk threads of length $\ell$ and carry similar charges $+q$ and $+q$.
(a) Prove that when the system is in equilibrium, separation between the balls is $x={\left(\frac{q^2\ell }{2\pi {\epsilon }_{0}mg}\right)}^{1/3}$, assuming the angle made by the threads with the vertical line passing through the point of suspension is small.
(b) Specify the rate $\left(dq/dt\right)$ with which the charge from each sphere leaks off if their velocity of approach varies as $v=\frac{a}{\sqrt{x}}$, where $a$ is a constant.