1. A ball is dropped from a bridge 125 meter above a river. After the ball has been falling for two second, a second ball is thrown straight down after it. What must its initial velocity be so that both hit the water at the same time? $g=10 m / s^2$
1) $39 m / s$2) $55.5 m / s$3) $26.7 m / s$4) $19.8 m / s$
Hint: For first ball, $s=\frac{1}{2} gt^2$ $$ \begin{aligned} 125 & =\frac{1}{2} \times 10 \times t^2 \\ t & =5 s \end{aligned} $$ For second ball, $t_1=3 s$ $s_1=ut_1+\frac{1}{2} gt_1^2$ $125=3 u+\frac{1}{2} \times 10 \times 9$ $u=\frac{80}{3}=26.7 m / s$
2. A ball is dropped from a bridge 125 meter above a river. After the ball has been falling for two second, a second ball is thrown straight down after it. what is the velocity of the first ball when it hits the water? $g=10 m / s^2$
1) $50 m / s$2) $69 m / s$3) $55.5 m / s$4) $9.8 m / s$
Hint: v = gt = 10 x 5 = 50 m/s
3. A ball is dropped from a bridge 125 meter above a river. After the ball has been falling for two second, a second ball is thrown straight down after it. what is the velocity of the second ball? $g=10 m / s^2$
1) 56.66 m/s2) $45.5 m / s$3) $38.4 m / s$4) $19.8 m / s$
Hint: $v=u+gt=26.7+(10 \times 3)=26.7+30=56.7 m / s$
4. Two bodies of different masses $m_a$ and $m_b$ are dropped from two different heights, a and b . The ratio of time taken by the two to drop through these distances is
1) $a : b$2) $m_a / m_b : b / a$3) $\sqrt{a} : \sqrt{b}$4) $a^2 : b^2$
Hint: $s=\frac{1}{2} gt^2 \Rightarrow t=\sqrt{\frac{2 s}{g}} \Rightarrow t \propto \sqrt{s}$ $\frac{t_a}{t_b}=\frac{\sqrt{a}}{\sqrt{b}}$
5. A body is fired vertically upwards, with a velocity $60 m / s$. The time intervals between the instants when it has a vertical displacement of 120 m , is
1) 6 s2) 12 s3) $8 \sqrt{3} s$4) $4 \sqrt{3} s$
Hint: $h=ut-\frac{1}{2} gt^2$ $120=60 t-5 t^2$ $t^2-12 t+24=0$ $t^2-12 t=-24$ $t^2-12 t+36=-24+36$ $(t-6)^2=12$ $t=6 \pm \sqrt{12}$ $t_1=6-\sqrt{12} t_2=6+\sqrt{12}$ $\Delta t=t_2-t_1=2 \sqrt{12}=4 \sqrt{3} s$
6. A body dropped from the top of a tower covers a distance 7 x in the last second of its journey where x is the distance covered in first second. How much time does it take to reach the ground?
1) 3 s2) 4 s3) 5 s4) 6 s
Hint: In first second distance travelled, $x=\frac{1}{2} gt^2=\frac{1}{2} \times 10 \times(1)^2=5 m$ In last second, $7 x=35 m$ $s_n=u+\frac{g}{2}(2 n-1)$ $35=0+5(2 n-1)$ $2 n-1=7$ $n=4 s$
7. A body, fired vertically upwards with velocity u, attains the same speed after time T. At two instants, $t_1$ and $t_2$, it crosses a point in its path. The $t_1+t_2=$
1) T2) $\frac{3 T}{4}$3) $\frac{T}{2}$4) $\frac{5 T}{4}$
Hint: $h=ut-\frac{1}{2} gt^2$ $2 h=2 ut-gt^2$ $gt^2-2 ut+2 h=0$ $t_{1,2}=\frac{ \pm 2 u+\sqrt{4 u^2-4 g(2 h)}}{2 g}=\frac{2 u \pm \sqrt{4 u^2-8 gh}}{2 g}$ $t_1=\frac{2 u-\sqrt{4 u^2-8 gh}}{2 g}$ $t_2=\frac{2 u+\sqrt{4 u^2-8 gh}}{2 g}$ $t_1+t_2=2(\frac{2 u}{2 g})=\frac{2 u}{g}=T$
8. Two balls A and B of same masses are thrown from the top of the building. ' A ', thrown upwards with velocity v and 'B', thrown downwards with velocity v , then
1) Velocity of A is more than B at the ground.2) Velocity of B is more than A at the ground.3) Both A and B strike the ground with same velocity.4) None of these
Hint: $v^2=u^2+2 gh$ $\therefore v=\sqrt{u^2+2 gh}$ So. For both the cases, velocity will be equal.
9. An elevator car, whose floor to ceiling distance is equal to 2.7 m , starts ascending with constant acceleration of $1.2 ms^{-2}$. Two sec after the start, a bolt begins falling from the ceiling of the car. The free fall time of the bolt is
1) $\sqrt{0.54} s$2) $\sqrt{6} s$3) 0.7 s4) 1 s
Hint: $t=\sqrt{\frac{2 h}{(g+a)}}=\sqrt{\frac{2 \times 2.7}{(9.8+1.2)}}=\sqrt{\frac{5.4}{11}}=\sqrt{0.49}=0.7 s$ As $u=0$ and lift is moving upward with acceleration.
10. A ball is dropped from a bridge of 122.5 metre above a river. After the ball has been falling for two seconds, a second ball is thrown straight down after it. Initial velocity of second ball, so that both hit the water at the same time is
1) $49 m / s$2) $55.5 m / s$3) $26.1 m / s$4) $9.8 m / s$
Hint: For the first ball: $H=\frac{1}{2} gt^2$ $22.5=\frac{1}{2} gt^2$ For second ball: Time taken $=t-2 s$ $$ 122.5=u(t-2)+\frac{1}{2} g(t-2) 2 $$ $u(t-2)=2 g(t-1)$ From eq. (i), $t^2=\frac{2 \times 122.5}{9.8}=25$ $t=5 s$ From eq. (iii), $3 u=8 \times 9.8=78.4$ $u=\frac{78.4}{3}=26.1 m / s$
11. If the velocity of a body, thrown upwards, is increased by $10 \%$, the maximum height reached by it will increase by
1) 20%2) $21 \%$3) $10 \%$4) 11
Hint: v = u + 10%u= 11} $v \propto h$ - $\frac{h_2}{h_1}=(\frac{11}{10})^2=\frac{121}{100}$ $\frac{h_2-h_1}{h_1}\times100=(\frac{121-100}{100})\times100=21%$
12. Two balls A and B of same masses are thrown from the top of the building. ' A ', thrown upwards with velocity v and 'B', thrown downwards with velocity v , then
1) Velocity of A is more than B at the ground.2) Velocity of B is more than A at the ground.3) Both A and B strike the ground with same velocity.4) None of these
Hint: $v^2=u^2+2 gh$ $\therefore v=\sqrt{u^2+2 gh}$ So. For both the cases, velocity will be equal.
13. An elevator car, whose floor to ceiling distance is equal to 2.7 m , starts ascending with constant acceleration of $1.2 ms^{-2}$. Two sec after the start, a bolt begins falling from the ceiling of the car. The free fall time of the bolt is
1) $\sqrt{0.54} s$2) $\sqrt{6} s$3) 0.7 s4) 1 s
Hint: $t=\sqrt{\frac{2 h}{(g+a)}}=\sqrt{\frac{2 \times 2.7}{(9.8+1.2)}}=\sqrt{\frac{5.4}{11}}=\sqrt{0.49}=0.7 s$ As $u=0$ and lift is moving upward with acceleration.
14. A ball is dropped from a bridge of 122.5 metre above a river. After the ball has been falling for two seconds, a second ball is thrown straight down after it. Initial velocity of second ball, so that both hit the water at the same time is
1) $49 m / s$2) $55.5 m / s$3) $26.1 m / s$4) $9.8 m / s$
Hint: For the first ball: $H=\frac{1}{2} gt^2$ $22.5=\frac{1}{2} gt^2$ For second ball: Time taken $=t-2 s$ $$ 122.5=u(t-2)+\frac{1}{2} g(t-2) 2 $$ $u(t-2)=2 g(t-1)$ From eq. (i), $t^2=\frac{2 \times 122.5}{9.8}=25$ $t=5 s$ From eq. (iii), $3 u=8 \times 9.8=78.4$ $u=\frac{78.4}{3}=26.1 m / s$
15. A body thrown vertically up to reach its maximum height in t second. The total time from the time of projection to reach a point at half of its maximum height while returning (in second) is
1) $\sqrt{2} t$2) $(1+\frac{1}{\sqrt{2}}) t$3) $\frac{3 t}{2}$4) $\frac{t}{\sqrt{2}}$
Hint: Time required to reach maximum height $=t$ Time required to reach half of maximum height is $t_1$ For downwards motion, $h=\frac{1}{2} gt^2$ $h_1=\frac{1}{2} gt_1^2=\frac{h}{2}$ $\therefore t_1^2=\frac{t^2}{2} \Rightarrow t_1=\frac{t}{\sqrt{2}}$ Total time $=t+t_1=t+\frac{t}{\sqrt{2}}=(1+\frac{1}{\sqrt{2}}) t$
16. Two balls of different masses are thrown vertically upwards with the same speed. They pass through the point of projection in their downward motion, with
1) the heavier ball having more speed.2) the lighter ball having more speed.3) both having the same speed.4) can not be predicted as actual height achieved is not given.
Hint: Motion under gravity is independent of the mass of the body.
17. A body, of mass M , is thrown upwards with, from ground, with velocity u . The average acceleration of the body, during the time t in air is
1) $g / 2$2) g3) $3 g / 4$4) $2 g / 3$
Hint:
18. A body rolls down a stair case of 5 steps. Each step has height 0.1 m and width 0.1 m . With what velocity will the body reach the bottom?
1) $5 \sqrt{2} m / s$2) $\frac{1}{\sqrt{2}} m / s$3) $2 \sqrt{2} m / s$4) $\sqrt{\frac{5}{2}} m / s$
Hint: Horizontal distance covered by the body $=5 \times 0.1=0.5 m$ Vertical distance covered by the body $=5 \times 0.1=0.5 m$ If the time taken to hit the edge of $5^{th}$ step is t,then $ut=0.5 \Rightarrow t=\frac{0.5}{u}$ $\frac{1}{2} gt^2=0.5$ $\frac{1}{2} \times 10 \times \frac{0.25}{u^2}=0.5$ $u^2=\frac{10}{4}=\frac{5}{2}$ $u=\sqrt{\frac{5}{2}} m / s$
19. A player throws a ball vertically upwards with velocity u. At highest point,
1) Both the velocity and acceleration of the ball are zero.2) The velocity of the ball is u but its acceleration is zero.3) The velocity of the ball is zero but its acceleration is g.4) The velocity of the ball is u but its acceleration is g.
Hint: At the highest point velocity of the ball becomes zero, but its acceleration is equal to g.
20. A body, fired vertically upwards, reaches a maximum height H in time T . The distance it covers in time $3 T / 2$ is
1) $\frac{3 H}{2}$2) $\frac{7 H}{4}$3) $\frac{3 H}{4}$4) $\frac{5 H}{4}$
Hint: $H=\frac{1}{2} gT^2$ $t=T+\frac{T}{2}=\frac{3 T}{2}$ $h=\frac{1}{2} g(\frac{T}{2})^2=\frac{1}{2}(\frac{1}{2} gT^2)=\frac{H}{4}$ Total distance $=H+h=H+\frac{H}{4}=\frac{5 H}{4}$
21. When a ball is thrown up vertically 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) $3 v_0$3) $9 v_0$4) $\frac{3 v_0}{2}$
Hint: For maximum height, $u^2=2 gH_{max} \Rightarrow u \propto \sqrt{H_{max}}$ To triple the maximum height, ball should be thrown with velocity $\sqrt{3} v 0$.
22. Water drops fall from the roof of a building 20 m high at regular intervals of time. The first drop reaching the ground at the same instant fifth drop starts its fall. The time between two successive drops is $g=10 m / s^2$
1) 2 s2) 1 s3) 0.5 s4) 1.5 s
Hint: Time taken by first drop to reach the ground $t'=\sqrt{\frac{2 s}{g}}=\sqrt{\frac{2 \times 20}{10}}=2 s$ Time interval between successive drops, $t=\frac{t'}{4}=\frac{2}{4}=0.5 s$
23. If an iron ball and a wooden ball of the same radius are released from a height $h$ in vacuum, then time taken by both of them to reach ground will be
1) unequal.2) exactly equal.3) roughly equal4) zero.
Hint: The time taken by the ball to reach the ground from height h is given by, $t=\sqrt{\frac{2 h}{g}}$ It is independent of the mass of the ball.
24. If the velocity of a body, thrown upwards, is increased by $10 \%$, the maximum height reached by it will increase by
1) $20 \%$2) $21 \%$3) $10 \%$4) $11 \%$
Hint: $v=u+10 \% u=\frac{11}{10} u$ $v^2 \propto h$ $\frac{h_2}{h_1}=(\frac{v}{u})^2=(\frac{11}{10})^2=\frac{121}{100}$ $\frac{h_2-h_1}{h_1} \times 100=(\frac{121-100}{100}) \times 100=21 \%$
25. A body is dropped from the top of a tower, 30 m high. At the same time a body is fired from the ground, vertically upward, along the same line, with a speed of $10 m / s$. The time after which they will collide with each other is
1) $0.5 s$2) $1.5 s$3) 2 s4) 3 s
Hint: Two bodies are collided at height $h$ after time $t$. $$ h=10 t-\frac{1}{2} gt^2 $$ $$ H-h=\frac{1}{2} gt^2 $$ From eqs. (i) and (ii), $$ H-10 t+\frac{1}{2}{g t_{2}}=\frac{1}{2}{g t^{2}}^{2} $$ $$ t=\frac{H}{10}=\frac{30}{10}=3 s $$
26. A ball thrown up from the ground is observed at a certain height after times 4 s and 6 s . If the acceleration due to gravity is $10 ms^{-2}$, the velocity of throw is
1) $50 ms^{-1}$2) $100 ms^{-1}$3) $25 ms^{-1}$4) $75 ms^{-1}$
Hint: Time taken by the ball to reach upto highest point $t=4+(\frac{6-4}{2})=5 s$ $v=u+gt=0+(10 \times 5)=50 m / s$