Gravitational Field Due to a Disc

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 $h_0$ on a horizontal floor and keeps rebounding after hitting the floor. Find: (i) speed after $n$ rebounds

(ii) Total distance travelled by the ball before stopping

(iii) Height it rises after nth rebound

(iv) Total time before stopping

(coefficient of restitution is $e$)

1. (i) $e^n\sqrt{2g{h}_0}$, (ii)$e^{2n}{h}_0$, (iii) $h_0\left[\frac{1+e^2}{1-e^2}\right]$, (iv) $\left(\frac{1+e}{1-e}\right)\sqrt{\frac{2h_0}{g}}$

2. (i) $e^{2n}\sqrt{2g{h}_0}$, (ii)$e^{4n}{h}_0$, (iii) $h_0\left[\frac{1-e^2}{1+e^2}\right]$, (iv) $\left(\frac{1+e}{1-e}\right)\sqrt{\frac{2h_0}{g}}$

3. (i)$e^n\sqrt{2g{h}_0}$, (ii)$e^{-4n}{h}_0$, (iii)infinite (iv) infinite

4. None of these

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 $\frac{3}{4}$, after what time interval the ball will strike the plane second time :

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

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

Q. A ball is dropped from a height of $30\text{m}$ in downward direction on the stationary floor. If the value of coefficient of restitution is $0.6$ and ball rebounds back up to the height $\left(h^{\prime }\right)$, find the height $\left(h^{\prime }\right)$ after first collision.
(Take $g=10{\text{m/s}}^2$)

1. $21.6\text{m}$
2. $22.8\text{m}$
3. $10.8\text{m}$
4. $26.8\text{m}$

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

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. 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. $\text{None of these}$