Problem Set

Q. Three particles $A,B$ and $C$ are situated at the vertices of an equilateral triangle $ABC$ of side $d$ at $t=0$. Each of the particles moves with constant speed $v$. $A$ always has its velocity along $AB$ i.e. along the line joining the two particles, $B$ along $BC$ and $C$ along $CA$. At what time will the particles meet each other?

1. $t=\frac{d}{3v}$
2. $t=\frac{2d}{3v}$
3. $t=\frac{2d}{v}$
4. $t=\frac{d}{v}$

Q. A stone is dropped from a height $h$. Another stone is thrown up simultaneously from the ground, which can reach to maximum height of $4h$. The two stones will cross each other after time

1. $\sqrt{\frac{h}{8g}}$
2. $\sqrt{2gh}$
3. $\sqrt{8gh}$
4. $\sqrt{\frac{h}{2g}}$

Q. A monkey is descending from the branch of a tree with constant acceleration. If the breaking strength of branch is $75\%$ of the weight of the monkey, the minimum acceleration with which monkey can slide down without breaking the branch is

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

Q. The acceleration (in $m/s^2$) of movable pulley P is and block B is , if acceleration of block $A=1{\text{m/s}}^2$ downwards.

Q. Two cars, initially at a separation of $12\text{m}$, start simultaneously. First car $A$, starting from rest moves with an acceleration $2{\text{m/s}}^2$ , whereas the car $B$, which is ahead, moves with a constant velocity $1\text{m/s}$, away from car $A$ along the same direction. Find the time when car $A$ overtakes car $B$.

1. $4\text{s}$
2. $6\text{s}$
3. $5\text{s}$
4. $7\text{s}$

Q. A stone is dropped from the top of a $400\text{m}$ high tower. At the same time another stone is projected vertically upwards from the ground with a speed of $50\text{m/s}$. The two stones will cross each other after a time

1. $2\text{s}$
2. $4\text{s}$
3. $6\text{s}$
4. $8\text{s}$

Q. A particle is moving along the path given by $y=\frac{C{t}^6}{6}$ where $C$ is a positive constant. The relation between acceleration $\left(a\right)$ and the velocity of the particle at $t=5s$ is given by $a=\eta v$, then the value of $5\eta$ is

Q. In a car race, car $A$ takes $20$ seconds less than car $B$ to finish and passes the finishing point with speed $v$ more than that of car $B$. Assuming that both cars start from rest and travel with a constant acceleration of $50{\text{m/s}}^2$ and $40{\text{m/s}}^2$ respectively, what is the value of $v$ ?

1. $894\text{m/s}$
2. $620\text{m/s}$
3. $682\text{m/s}$
4. $864\text{m/s}$

Q. A stone is dropped from a height $h$. Another stone is thrown up simultaneously from the ground, which can reach to maximum height of $4h$. The two stones will cross each other after time

1. $\sqrt{\frac{h}{8g}}$
2. $\sqrt{8gh}$
3. $\sqrt{2gh}$
4. $\sqrt{\frac{h}{2g}}$

Q. A car has a headlight which can illuminate a horizontal straight road in front upto a distance of $10\text{m}$. If the coefficient of friction between tyre and road is $0.5$, the maximum safe speed of the car during a night drive is
[Neglect the reaction time of the driver and take $g=10{\text{m/s}}^2$]

1. $5\text{m/s}$
2. $8\text{m/s}$
3. $10\text{m/s}$
4. $12\text{m/s}$

Q. A body is dropped from the top of a tower covers $\frac{7}{16}$ of total height isn the last second of its fall. The time of fall is

1. $2s$
2. $4s$
3. $<1s$
4. $>5s$

Q. A ball is projected vertically upwards with a velocity of 40 m/s from the top of a cliff 100 m high .Find the total time taken the ground.

1. 4 sec
2. 8 sec
3. 15.8 sec
4. 12 sec

Q. In the arrangement shown in $Fig.6.337$ at a particular instant, the roller is coming down with a speed of $12m{s}^{-1}$ and C is moving up with $4m{s}^{-1}$, At the same instant, it is also known that w.r.t. pulley P, block A is moving down with speed $3m{s}^{-1}$. Determine the motion of block B (velocity) w.r.t. ground

1. $4m{s}^{-1}$ in downward direction
2. $7M{s}^{-1}$ in upward direction
3. $3m{s}^{-1}$ in upward direction
4. $7m{s}^{-1}$ in downward direction

Q. Three particles $A,B,C$ are situated at the vertices of an equilateral triangle of side $l$. Each of the particle starts moving with a constant velocity $v$ such that $A$ is always directed towards $B$, $B$ towards $C$ and $C$ towards $A$. Find the time when they meet.

1. $\frac{2l}{v}$
2. none
3. $\frac{2l}{3v}$
4. $\frac{l}{\sqrt{3}v}$

Q. When the velocity of body is variable, then

1. Its speed may be constant
2. Its acceleration may be constant
3. Its average acceleration may be constant
4. All of these

Q. Three particles $A,B$ and $C$ are situated at the vertices of an equilateral triangle $ABC$ of side $d$ at $t=0$. Each of the particles moves with constant speed $v$. $A$ always has its velocity along $AB$ i.e. along the line joining the two particles, $B$ along $BC$ and $C$ along $CA$. At what time will the particles meet each other?

1. $t=\frac{d}{3v}$
2. $t=\frac{2d}{3v}$
3. $t=\frac{2d}{v}$
4. $t=\frac{d}{v}$

Q. A stone is dropped from the top of a $400\text{m}$ high tower. At the same time another stone is projected vertically upwards from the ground with a speed of $50\text{m/s}$. The two stones will cross each other after a time

1. $2\text{s}$
2. $4\text{s}$
3. $6\text{s}$
4. $8\text{s}$

Q. A stone is dropped from a certain height which can reach the ground in $5s$. If the stone is stopped after $3s$ of its fall and then allowed to fall again, then the time taken by the stone to reach the ground for the remaining distance is

1. $3s$
2. $4s$
3. $2s$
4. $8s$

Q. In an experiment with a beam balance, an unknown mass $m$ is balanced by two known masses of $16kg$ and $4kg$ as shown in fig. Find $m$.

1. $10kg$
2. $12kg$
3. $6kg$
4. $8kg$

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

Q. A body of mass m falls freely through a height h from the top of a tower. The velocity just before touching the ground is $\sqrt{\frac{3}{2}gh}$ . The air drag is:

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

Q. The acceleration of a particle is increasing linearly with time $t$ as $bt$. The particle starts from the origin with an initial velocity $v_0$. The distance travelled by the particle in time $t$ will be

1. $v_{0}t+\frac{1}{3}b{t}^2$
2. $v_{0}t+\frac{1}{3}b{t}^3$
3. $v_{0}t+\frac{1}{2}b{t}^2$
4. $v_{0}t+\frac{1}{6}b{t}^3$

Q. Position of a particle moving in $x-y$ plane as function of time t $2ti+4t^{2}j$. Equation of trajectory of the particle is

1. $y=x^2$
2. $y=2x$
3. $y^2=x$
4. $y=x$

Q. Which of the following relations is wrong ?

1. $\overline{a}=\overline{r}\times \overline{\alpha }$
2. $\overline{J}=\overline{r}\times \overline{P}$
3. $\overline{v}=\overline{\omega }\times \overline{r}$
4. $\tau =\frac{d\overline{J}}{dt}$

Q. The acceleration (in $m/s^2$) of movable pulley P is and block B is , if acceleration of block $A=1{\text{m/s}}^2$ downwards.

Q. Two cars, initially at a separation of $12\text{m}$, start simultaneously. First car $A$, starting from rest moves with an acceleration $2{\text{m/s}}^2$ , whereas the car $B$, which is ahead, moves with a constant velocity $1\text{m/s}$, away from car $A$ along the same direction. Find the time when car $A$ overtakes car $B$.

1. $4\text{s}$
2. $6\text{s}$
3. $5\text{s}$
4. $7\text{s}$

Q. An object is allowed to fall freely from a tower of height 39.2 m ; In exactly at the same time another stone is thrown from the bottom of the in tower in vertically upward direction with a velocity of 19.6 $ms-1$ Calculate when and where these two stones would meet ?

1. $3s,19.6m$
2. $2s,19.6m$
3. $1.5s.19.6m$
4. $4s,19.6m$

Q. A particle is projected from ground with a speed of $v_0=80m/s$ at angle of ${60}^{\circ }$ with horizontal. During first 3 seconds average velocity is $\overrightarrow{v}$ & acceleration is $\overrightarrow{a}$ & displacement is $\overrightarrow{\Delta r}$, then $(\overrightarrow{a}\times \overrightarrow{v}).\overrightarrow{\Delta r}$ equals to

1. $20units$
2. $36units$
3. $40\sqrt{3}units$
4. $Zero$

Q.

A body is projected from the ground with the velocity of $10{ms}^{-1}$ such that it makes an angle of 30o\displaystyle { 30 }^{ o }30o with the horizontal. What is the horizontal velocity at the maximum height?

1. 62ms−1\displaystyle 6\sqrt { 2 } \quad { ms }^{ -1 }62​ms−1
2. 92ms−1\displaystyle 9\sqrt { 2 } \quad { ms }^{ -1 }92​ms−1
3. 35ms−1\displaystyle 3\sqrt { 5 } \quad { ms }^{ -1 }35​ms−1
4. 53ms−1\displaystyle 5\sqrt { 3 } \quad { ms }^{ -1 }53​ms−1