Applications of equations of motion

Q. A ball is thrown vertically downward with a velocity of $20\text{m/s}$ from the top of a tower. It hits the ground after some time with a velocity of $80\text{m/s}$. The height of the tower is :
$(g=10{\text{m/s}}^2)$

1. $320\text{m}$
2. $300\text{m}$
3. $360\text{m}$
4. $340\text{m}$

Q.

A rubber ball is released from a height of $5m$ above the floor. It bounces back repeatedly, always rising to $\frac{81}{100}$ of the height through which it falls. Find the average speed of the ball. (Take $g=10m{s}^{-2}$)

1. $2.5m{s}^{-1}$

2. $3.5m{s}^{-1}$

3. $3.0m{s}^{-1}$

4. $2.0m{s}^{-1}$

Q. An engine of a train, moving with uniform acceleration, passes the signal post with velocity $u$ and the last compartment passes the signal post with velocity $v$. The velocity with which the middle point of the train passes the signal post is :

1. $\sqrt{\frac{v^2-u^2}{2}}$
2. $\sqrt{\frac{v-u}{2}}$
3. $\sqrt{\frac{v^2+u^2}{2}}$
4. $\sqrt{\frac{v+u}{2}}$

Q. In a car race on straight road, car $\text{A}$ takes time $t$ less than car $\text{B}$ at the finish and passes finishing point with a speed $v$ more than that of car $\text{B}$. Both the cars start from rest and travel with constant acceleration $\alpha$ and $\beta$ respectively. Then $v$ is equal

1. $\left(\sqrt{2\alpha \beta }\right)t$
   
2. $\left(\sqrt{\alpha \beta }\right)t$
   
3. $\left(\frac{\alpha +\beta }{2}\right)t$
4. $\left(\frac{2\alpha +\beta }{\alpha +\beta }\right)t$

Q. A particle starts from a point with a velocity of $+6\text{m/s}$ and moves with an acceleration of $-2{\text{m/s}}^2$. After what time will the particle be at the starting point?

1. $2\text{s}$
2. $6\text{s}$
3. $4\text{s}$
4. $5\text{s}$

Q. Two balls are dropped from different heights at different instants. Second ball is dropped $2\text{s}$ after the first ball. If both the balls reach the ground simultaneously, after $5\text{s}$ of dropping the first ball, then the difference between the initial heights of the two balls will be
$(\text{Take}g=9.8{\text{m/s}}^2)$

1. $58.8\text{m}$
2. $78.4\text{m}$
3. $98.0\text{m}$
4. $117.6\text{m}$

Q. A car and a truck move in the same straight line at the same instant of time from the same point. The car moves with a constant velocity of $40\text{m/s}$ and truck starts with a constant acceleration of $4{\text{m/s}}^2$. Find the time $t$ that elapses before the truck catches with the car and also find the greatest distance $S_{max}$ between them prior to it

1. $t=5\text{sec},S_{max}=100\text{m}$
2. $t=20\text{sec},S_{max}=200\text{m}$
3. $t=15\text{sec},S_{max}=50\text{m}$
4. $t=8\text{sec},S_{max}=100\text{m}$

Q. A ball is thrown vertically upwards with a velocity $u$ from the ground. The ball attains a maximum height $H_{max.}$. Then find out the displacement at which ball has half of the maximum speed.

1. $\frac{H_{max}}{2}$
2. $\frac{3}{2}{H}_{max}$
3. $$\frac{2}{3}{H}_{max}$$
4. $$\frac{3}{4}{H}_{max}$$

Q.

A bullet when first into a target loses half of its velocity after penetrating $20cm$. Further distance of penetration before it comes to rest is

1. $6.66cm$

2. $3.33cm$

3. $12.5cm$

4. $10cm$

5. $5cm$

Q. A particle starts from a point with a velocity of $+6\text{m/s}$ and moves with an acceleration of $-2{\text{m/s}}^2$. After what time will the particle be at the starting point?

1. $2\text{s}$
2. $6\text{s}$
3. $4\text{s}$
4. $5\text{s}$

Q. A ball thrown vertically upwards with a speed of $19.6{\text{ms}}^{-1}$ from the top of a tower returns to the earth in $6\text{s}$. Find the height of the tower.
(Take $g=9.8{\text{m/s}}^2$)

1. $27.8\text{m}$
2. $58.8\text{m}$
3. $70\text{m}$
4. $92\text{m}$

Q. A girl throws a ball vertically upwards with an initial speed of $15m/s$. The ball was released when it was at $2.0m$ above the ground. The girl catches it at the same point as the point of projection. What is the maximum height reached by the ball?

1. $11.5m$
2. $13.5m$
3. $15.5m$
4. $17.5m$

Q. A car accelerates uniformly from $13{\text{ms}}^{–1}$ to $31{\text{ms}}^{–1}$ while entering the motorway, covering a distance of $220\text{m}$. Then the acceleration of the car will be:

1. $2.9{\text{ms}}^{-2}$
2. $1.8{\text{ms}}^{-2}$
3. $4{\text{ms}}^{-2}$
4. $2.2{\text{ms}}^{-2}$

Q. A ball is thrown vertically downward with a velocity of $20\text{m/s}$ from the top of a tower. It hits the ground after some time with a velocity of $80\text{m/s}$. The height of the tower is :
$(g=10{\text{m/s}}^2)$

1. $320\text{m}$
2. $300\text{m}$
3. $360\text{m}$
4. $340\text{m}$

Q. A body starts from rest and moving with an acceleration of $1{\text{ms}}^{–2}$. The displacement of the body in $5$ seconds is

1. $12.5\text{m}$
2. $25\text{m}$
3. $7.5\text{m}$
4. $15\text{m}$

Q. An elevator car whose floor to ceiling distance is equal to 2 m starts ascending with constant acceleration 2 $m/s^2$. $5sec$ after the start, a bolt begins falling from ceiling of car. Time after which bolt hits floor of elevator starting from fall. $(g=10m/s^2)$

1. $0.7$ s
2. $\frac{1}{\sqrt{2}}$ s
3. $1.73$ s
4. $\frac{1}{\sqrt{3}}$ s

Q. A ball is thrown vertically upwards with a speed $v$ from a height $h$ metre above the ground. The time taken for the ball to hit ground is

1. $\frac{v}{g}\sqrt{1-\frac{2hg}{v^2}}$
2. $\frac{v}{g}\sqrt{1+\frac{2hg}{v^2}}$
3. $\sqrt{1+\frac{2hg}{v^2}}$
4. $\frac{v}{g}[1+\sqrt{1+\frac{2hg}{v^2}}]$

Q. When a ball is thrown vertically upwards with velocity $v_0$, it reaches a maximum height of $h$. If one wishes to triple the maximum height then the ball should be thrown with velocity.

1. $\sqrt{3}{v}_0$
2. $3v_0$
3. $9v_0$
4. $\frac{3}{2}{v}_0$

Q. A ball is thrown vertically upwards with a velocity $u$ from the ground. The ball attains a maximum height $H_{max.}$. Then find out the displacement at which ball has half of the maximum speed.

1. $\frac{H_{max}}{2}$
2. $\frac{3}{2}{H}_{max}$
3. $$\frac{2}{3}{H}_{max}$$
4. $$\frac{3}{4}{H}_{max}$$

Q.

A body starts from rest and moves with a uniform acceleration. Find the ratio of the distance covered in the ${\mathrm{n}}^{\text{th}}$ sec to the distance covered in $nsec$.

Q. Two bodies A (of mass $1\text{kg})$ and B (of mass $3\text{kg})$ are dropped from heights of $16\text{m}$ and $25\text{m}$, respectively. The ratio of the time taken by them to reach the ground is

1. $\frac{4}{5}$
2. $\frac{5}{4}$
3. $\frac{12}{5}$
4. $\frac{5}{12}$

Q. A ball is thrown vertically upwards with velocity of $10\text{m/s}$ from the top of a tower. It returns to ground after some time with speed of $60\text{m/s}$. The height of the tower is

$(g=10{\text{m/s}}^2)$

1. $375\text{m}$
2. $175\text{m}$
3. $125\text{m}$
4. $225\text{m}$

Q. A particle is thrown vertically upwards. If its velocity at half of the maximum height is $10\frac{m}{s}$, then maximum height attained by it is (Take $g=10\frac{m}{s^2}$)

1. $15m$
2. $20m$
3. $10m$
4. $5m$

Q. A stone is dropped from a building of height $h$ and it reaches after $t$ seconds on earth. From the same building if two stones are thrown, one upwards and other downwards, with the same velocity $u$ and they reach the earth surface after $t_1$ and $t_2$ seconds respectively, then

1. $t=t_1-t_2$
   
2. $t=\frac{t_1+t_2}{2}$
   
3. $t=\sqrt{t_1{t}_2}$
   
4. $t=t_1^2{t}_2^2$
   

Q. Some parameters are given in Column - I which are to be matched with their correct answers given in Column - II.
Choose the correct option.
$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{a)}& \text{Velocity at}t=0\text{of a particle following}& \text{p)}& \alpha \\ & \text{equation}x=u(t-8)+\alpha (t-2{)}^2& & \\ \text{b)}& \text{Acceleration of a particle moving according to equation}& \text{q)}& 2\alpha \\ & x=4(t-5)+\alpha (t-2{)}^2\text{at}t=0& & \\ \text{c)}& \text{Velocity of the particle moving according to equation}& \text{r)}& u\\ & x=ut+5\alpha t^{2}ln(1+\frac{t}{2})+\alpha t^2\text{at}t=0& & \\ \text{d)}& \text{Acceleration of the particle moving according to the equation}& \text{s)}& u-4\alpha \\ & x=ut+5\alpha t^{2}ln(1+\frac{t}{2})+\alpha t^2\text{at}t=0& & \end{array}$

1. $a\to sb\to qc\to rd\to q$
2. $a\to pb\to rc\to qd\to q$
3. $a\to pb\to rc\to sd\to q$
4. $a\to pb\to qc\to rd\to s$

Q. A stone is dropped from a balloon going up with a uniform velocity of $5.0\text{m/s}$. If the balloon was $50\text{m}$ high when the stone was dropped, find its height when stone hits the ground.

1. $50\text{m}$
2. $68.5\text{m}$
3. $78.5\text{m}$
4. $40\text{m}$

Q. A particle is thrown vertically upwards. It's velocity at half of the height is $10\text{m/s}$, then the maximum height attained by it will be:

1. $10\text{m}$
2. $20\text{m}$
3. $15\text{m}$
4. $25\text{m}$

Q. A particle is dropped under gravity from rest from a height $h(g=9.8\frac{m}{se{c}^2})$ and it travels a distance $\frac{9h}{25}$ in the last second, the height h is

1. 100 m
2. 122.5 m
3. 145 m
4. 167.5 m

Q. A ball $A$ is dropped from a $44.1\text{m}$ high cliff. Two seconds later, another ball $B$ is thrown downward from the same place with some initial speed. The two balls reach the ground together. Find the speed with which the ball $B$ was thrown.
$[\text{Take}g=9.8{\text{m/s}}^2]$

1. $39.2\text{m/s}$
2. $44.1\text{m/s}$
3. $24.6\text{m/s}$
4. $12.6\text{m/s}$

Q. From the top of a tall mountain, a stone is dropped. One second later, another stone is projected downwards with velocity $20{\text{ms}}^{-1}$. Find the distance from top where both stones meet.

1. $\frac{45}{4}\text{m}$
2. $40\text{m}$
3. $10\text{m}$
4. $\frac{50}{8}\text{m}$