Arihant AIEEE Physics

D.B. Singh

Chapter 3

Kinematics - all with Video Answers

Educators

Chapter Questions

Problem 1

Mark correct option or options:
(a) displacement may be equal to the distance
(b) displacement must be in the direction of the acceleration of the body
(c) displacement must not be in the direction of velocity
(d) none of the above

Mahendra K

Mahendra K

Numerade Educator

Problem 2

In the two dimensional motions:
(a) $x-t$ graph gives actual path of the particle
(b) $y$ -t graph gives actual path of the particle
(c) $\sqrt{x^{2}+y^{2}}$ versus $t$ graph gives the actual path of the particle
(d) $y-x$ graph gives actual path of particle

Mahendra K

Numerade Educator

Problem 3

A cat wants to catch a rat. The cat follows the path whose equation is $x+y=0 .$ But rat follows the path whose equation is $x^{2}+y^{2}=4$. The co-ordinates of possible points of catching the rat are:
(a) $(\sqrt{2}, \sqrt{2})$
(b) $(-\sqrt{2}, \sqrt{2})$
(c) $(\sqrt{2}, \sqrt{3})$
(d) $(0,0)$

Mahendra K

Numerade Educator

Problem 4

A deer wants to save her life from a lion. The lion follows a path whose equation is $x^{2}+y^{2}=16$. For saving life, the deer moves on a path whose equation is/are:
(a) $x^{2}+y^{2}=4$
(b) $x^{2}+y^{2}=16$
(c) $x^{2}+y^{2}-64=0$
(d) both (a) and (c) are correct

Mahendra K

Numerade Educator

Problem 5

Which of the following position-time graph does not exist in nature?

Mahendra K

Numerade Educator

Problem 6

There is a square caromboard of side $a$. A striker is projected in hole after two successive collisions. Assuming the collisions to be perfectly elastic and the surface to be smooth. The angle of projection of striker is :
(a) $\cot ^{-1}\left(\frac{3}{4}\right)$
(b) $\cos ^{-1}\left(\frac{3}{4}\right)$
(c) $\sin ^{-1}\left(\frac{3}{4}\right)$
(d) none of these

Mahendra K

Numerade Educator

Problem 7

Speed is to velocity as:
(a) centimetre is to metre
(b) force is to torque
(c) velocity is to acceleration
(d) distance is to displacement

Mahendra K

Numerade Educator

Problem 8

A person travelling on a straight line moves with a uniform velocity $v_{1}$ for some time and with uniform velocity $v_{2}$ for the next the equal time. The average velocity $v$ is given by:
(a) $v=\frac{v_{1}+v_{2}}{2}$
(b) $\frac{2}{v}=\frac{1}{v_{1}}+\frac{1}{v_{2}}$
(c) $v=\sqrt{v_{1} v_{2}}$
(d) $\frac{1}{v}=\frac{1}{v_{1}}+\frac{1}{v_{2}}$

Mahendra K

Numerade Educator

Problem 9

A car moves at $80 \mathrm{~km} / \mathrm{hr}$ in the first half of total time of motion and at $40 \mathrm{~km} / \mathrm{h}^{-1}$ in the later half. Its average speed is:
(a) $60 \mathrm{~km} / \mathrm{hr}$
(b) $30 \mathrm{~km} / \mathrm{hr}$
(c) $120 \mathrm{~km} / \mathrm{hr}$
(d) none of these

Mahendra K

Numerade Educator

Problem 10

A particle moves with constant speed $v$ along a regular hexagon $A B C D E F$ in same order. (i.e., $A$ to $B, B$ to $C, C$ to $D, D$ to $E, E$ to $F$ and $F$ to $A$ ) The magnitude of average velocity for its motion from $A$ to $C$ is :
(a) $\underline{v}$
(b) $\frac{v}{2}$
(c) $\frac{\sqrt{3} v}{2}$
(d) none of these

Mahendra K

Numerade Educator

Problem 11

One rickshaw leaves Patna Junction for Gandhi Maidan at every 10 minute. The distance between Gandhi Maidan and Patna Junction is $6 \mathrm{~km}$. The rickshaw travels at the speed of $6 \mathrm{~km} / \mathrm{hr}$. What is the number of rickshaw that a rickshaw puller driving from Gandhi Maidan to Patna Junction must be in the route if he starts from Gandhi Maidan simultaneously with one of the rickshaw leaving Patna Junction:
(a) 11
(b) 12
(c) 5
(d) 1

Mahendra K

Numerade Educator

Problem 12

During the shooting of a super hit film 'MARD' Amitabh Bachchan was waiting for his beloved 'Amrita Singh' with his dog. When he saw her approaching, the dog was excited and dashed to her then back to master and so on, never stopping. How far would you estimate the dog ran if its speed is $30 \mathrm{~km} / \mathrm{hr}$ and each of them walked at $4 \mathrm{~km} / \mathrm{hr}$, starting from a distance $400 \mathrm{~m}$ apart?
(a) $400 \mathrm{~m}$
(b) $880 \mathrm{~m}$
(c) $1500 \mathrm{~m}$
(d) $30 \mathrm{~km}$

Mahendra K

Numerade Educator

Problem 13

Two particles start from the same point with different speeds but one moves along $y=a \sin \omega x$ and other moves along curve $y=a \cos \omega x:$
(a) they must collide after some time
(b) they never collide with each other
(c) they may collide at a point $P\left(\frac{\pi}{4 \omega}, \frac{a}{\sqrt{3}}\right)$

Mahendra K

Numerade Educator

Problem 14

A sheet of wood moves over a smooth surface (shown in the figure). The magnitude of velocity of $C$ is :
(a) $v$
(b) $2 v \cos \theta$
(c) $2 v \sin \theta$
(d) $2 v$

Mahendra K

Numerade Educator

Problem 15

The given hing construction consists of two rhombus with the ratio $3: 2$. The vertex $A_{2}$ moves in the horizontal direction with a velocity $v .$ The velocity of $A_{1}$ is :
(a) $0.6 v$
(b) $0.7 v$
(c) $3 v$
(d) $2 v$

Mahendra K

Numerade Educator

Problem 16

In the arrangement shown in figure, the ends $P$ and $Q$ of an inextensible string move downwards with uniform speed $u$. Pulleys $A$ and $B$ are fixed. The mass $m$ moves upwards with a speed:
(a) $2 u \cos \theta$
(b) $\frac{u}{\cos \theta}$
(c) $\frac{2 u}{\cos \theta}$
(d) $u \cos \theta$

Mahendra K

Numerade Educator

Problem 17

In the given figure, find the speed of pulley $P$ :
(a) $\frac{v}{2}$
(b) $2 v \cos \theta$
(c) $\frac{2 v}{\cos \theta}$
(d) $\frac{v}{2 \sin \theta}$

Mahendra K

Numerade Educator

Problem 18

A tractor $A$ is used to hoist the body $B$ with the pulley arrangement shown in fig. If $A$ has a forward velocity $v_{A}$. find the velocity of the body $B:$
(a) $\frac{x w_{A}}{2}$
(b) $\frac{x p_{A}}{d}$
(c) $\frac{x v_{A}}{2 \sqrt{x^{2}+d^{2}}}$
(d) none of these

Supratim Pal

Numerade Educator

Problem 19

A link $A B$ is moving in a vertical plane. At a certain instant when the link is inclined $60^{\circ}$ to the horizontal, the point $A$ is moving horizontally at $3 \mathrm{~m} / \mathrm{s}$, while $B$ is moving in the vertical direction. What is the velocity of $B$ ?
(a) $\frac{1}{\sqrt{3}} \mathrm{~m} / \mathrm{s}$
(b) $2 \sqrt{3} \mathrm{~m} / \mathrm{s}$
(c) $\sqrt{3} \mathrm{~m} / \mathrm{s}$
(d) $\frac{\sqrt{3}}{2} \mathrm{~m} / \mathrm{s}$

Mahendra K

Numerade Educator

Problem 20

Two intersecting straight lines moves parallel to themselves with speeds $3 \mathrm{~m} / \mathrm{s}$ and $4 \mathrm{~m} / \mathrm{s}$ respectively. The speed of the point of intersection of the lines, if the angle between them is $90^{\circ}$ will be:
(a) $5 \mathrm{~m} / \mathrm{s}$
(b) $3 \mathrm{~m} / \mathrm{s}$
(c) $4 \mathrm{~m} / \mathrm{s}$
(d) none of these

Mahendra K

Numerade Educator

Problem 21

The displacement time graph is shown in figure. The instantaneous velocity is negative ac the point:
(a) $\bar{D}$
(b) $F$
(c) $\mathrm{C}$
(d) $E$

Mahendra K

Numerade Educator

Problem 22

In the given $x$ -t curve:
(a) the velocity at $A$ is zero but at $B$ is non-zero
(b) the velocity at $A$ and $B$ is zero $x$
(c) the velocity at $A$ and $B$ is non-zero
(d) the directions of velocity at $A$ and $B$ are definite

Mahendra K

Numerade Educator

Problem 23

A particle moves along $X$ -axis whose velocity varies with time as shown in the figure:

Mahendra K

Numerade Educator

Problem 24

The position of a particle at any instant $t$ is given by $x=a$ cos $\omega t .$ The speed-time graph of the particle is:

Mahendra K

Numerade Educator

Problem 25

Which of the following speed-time graphs exists in the nature ?

Mahendra K

Numerade Educator

Problem 26

Two particles describe the same circle of radius $a$ in the same sense with the same speed $v$. What is their relative angular velocity?
(a) $v$ a
(b) $2 \mathrm{v} / a$
(c) $w / 2 a$
(d) va

Mahendra K

Numerade Educator

Problem 27

A particle is moving on a straight line path with constant acceleration directed along the direction of instantaneous velocity. Which of following statements are false about the motion of particle?
(a) Particle may reverse the direction of motion
(b) Distance covered is not equal to magnitude of displacement
(c) The magnitude of average velocity is less than average speed
(d) All the above

Mahendra K

Numerade Educator

Problem 28

Mark the correct statements for a particle going on a straight line:
(a) if the velocity and acceleration have opposite signs, the object is slowing down
(b) if the position and velocity have opposite signs, the particle is moving towards the origin
(c) if the velocity is zero at an instant, the acceleration should also be zero at that instant
(d) if the velocity is zero for a time interval, the acceleration is zero at any instant within the time interval
(e) $(\mathrm{a}),(\mathrm{b})$ and
(d) are correct.

Mahendra K

Numerade Educator

Problem 29

A particle of mass $m$ is initially situated at the point $P$ inside a hemispherical surface of radius
$r$ as shown in figure. $\mathrm{A}$ horizontal acceleration of magnitude $a_{0}$ is suddenly produced on the particle in the horizontal direction. If gravitational acceleration is neglected, the time taken by particle to touch the sphere again is:
(a) $\sqrt{\frac{4 r \sin \alpha}{a_{0}}}$
(3) $\sqrt{\frac{4 r \tan \alpha}{a_{0}}}$
(c) $\sqrt{\frac{4 r \cos \alpha}{a_{0}}}$
(d) none of these

Mahendra K

Numerade Educator

Problem 30

A particle starts with a velocity of $2 \mathrm{~m} / \mathrm{s}$ and moves in a straight line with a retardation of $0.1 \mathrm{~m} / \mathrm{s}^{2}$. The time that it takes to describe $15 \mathrm{~m}$ is :
(a) $10 \mathrm{~s}$ in its backward journey
(b) $30 \mathrm{~s}$ in its forward journey
(c) $10 \mathrm{~s}$ in its forward journey
(d) $30 \mathrm{~s}$ in its backward journey
(e) both (b) and (c) are correct

Mahendra K

Numerade Educator

Problem 31

A particle starts from rest with acceleration $2 \mathrm{~m} / \mathrm{s}^{2}$. The acceleration of the particle decreases down to zero uniformly during time-interval of 4 second. The velocity of particle after 2 second is :
(a) $3 \mathrm{~m} / \mathrm{s}$
(b) $4 \mathrm{~m} / \mathrm{s}$
(c) zero
(d) $8 \mathrm{~m} / \mathrm{s}$

Mahendra K

Numerade Educator

Problem 32

In the previous problem, the distance travelled by the particle during the time interval of $4 \mathrm{~s}$ is :
(a) $10.66 \mathrm{~m}$
(b) $20 \mathrm{~m}$
(c) $4 \mathrm{~m}$
(d) $2 \mathrm{~m}$

Mahendra K

Numerade Educator

Problem 33

If the greatest admissible acceleration or retardation of a train be 3 feet $/ \mathrm{sec}^{2}$, the least time taken from one station to another at a distance of 10 metre is [the maximum speed being 60 mile per hour] :
(a) $500 \mathrm{sec}$
(b) $58.67 \mathrm{sec}$
(c) $400 \mathrm{sec}$
(d) $314 \frac{2}{3} \mathrm{sec}$

Mahendra K

Numerade Educator

Problem 34

A person walks up a stalled escalator in 90 second. When standing on the same escalator, now moving, he is carried in 60 second. The time it would take him to walk up the moving escalator will be :
(a) $27 \mathrm{~s}$
(b) $72 \mathrm{~s}$
(c) $18 \mathrm{~s}$
(d) $36 \mathrm{~s}$

Mahendra K

Numerade Educator

Problem 35

A body starts from rest and moves with a constant acceleration. The ratio of distance covered in the $n$ th second to the distance covered in $n$ second is :
(a) $\frac{2}{n}-\frac{1}{n^{2}}$
(b) $\frac{1}{n^{2}}-\frac{1}{n}$
(c) $\frac{2}{n^{2}}-\frac{1}{n}$
(d) $\frac{2}{n}+\frac{1}{n^{2}}$

Mahendra K

Numerade Educator

Problem 36

A particle moving with a uniform acceleration along a straight line covers distances $a$ and $b$ in successive intervals of $p$ and $q$ second. The acceleration of the particle is :
(a) $\frac{p q(p+q)}{2(b p-a q)}$
(b) $\frac{2(a q-b p)}{p q(p-q)}$
(c) $\frac{b p-a q}{p q(p-q)}$
(d) $\frac{2(b p-a q)}{p q(p+q)}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 37

A body moves along $x$ -axis with velocity $v$. If the plot $v-x$ is an ellipse with major axis $2 A$ and minor axis $2 v_{0}$. the maximum acceleration has a modulus:
(a) $\frac{v_{0}^{2}}{A}$
(b) $\frac{A}{v_{0}^{2}}$
(c) $v_{0} A$
(d) none of these

Mahendra K

Numerade Educator

Problem 38

The distance time graph of a particle at time $t$ makes angle $45^{\circ}$ with respect to time axis. After one second, it makes angle $60^{\circ}$ with respect to time axis. What is the acceleration of the particle?
(a) $\sqrt{3}-1$ unit
(b) $\sqrt{3}+1$ unit
(c) $\sqrt{3}$ unit
(d) 1 unit

Mahendra K

Numerade Educator

Problem 39

The velocity-time plot for a particle moving on a straight line is shown in the figure, then:
(a) the particle has a constant acceleration
(b) the particle has never turned around
(c) the average speed in the interval 0 to $10 \mathrm{~s}$ is the same as the average speed in the interval $10 \mathrm{~s}$ to $20 \mathrm{~s}$
(d) both (a) and (c) are correct

Mahendra K

Numerade Educator

Problem 40

The acceleration of a train between two stations $2 \mathrm{~km}$ apart is shown in the figure. The maximum speed of the train is :
(a) $60 \mathrm{~m} / \mathrm{s}$
(b) $30 \mathrm{~m} / \mathrm{s}$
(c) $120 \mathrm{~m} / \mathrm{s}$
(d) $90 \mathrm{~m} / \mathrm{s}$

Mahendra K

Numerade Educator

Problem 41

When acceleration of a particle is $a=f(t)$, then:
(a) the velocity, starting from rest is $\int_{n}^{i} f(t) d t$
(b) velocity may be constant
(c) the velocity must not be function of time
(d) the speed may be constant with respect to time

Mahendra K

Numerade Educator

Problem 42

A particle moves in a straight line so that after $t$ second, the distance $x$ from a fixed point $O$ on the line is given by $x=(t-2)^{2}(t-5)$. Then:
(a) after $2 \mathrm{~s}$, velocity of particle is zero
(b) after $2 \mathrm{~s}$, the particle reaches at $\mathrm{O}$
(c) the acceleration is negative, when $t<3 \mathrm{~s}$
(d) all the above

Mahendra K

Numerade Educator

Problem 43

A bee flies in a line from a point $A$ to another point $B$ in 4 s with a velocity of $|t-2| \mathrm{m} / \mathrm{s}$. The distance between $A$ and $B$ in metre is :
(a) 2
(b) 4
(c) 6
(d) 8

Anand Jangid

Numerade Educator

Problem 44

When acceleration be function of velocity as $a=f(v)$. Then:
(a) the displacement $(x)=: \int \frac{\partial d v}{f(v)}$
(b) the acceleration may be constant
(c) the slope of acceleration versus velocity graph may be constant
(d) (a) and (c) are correct

Mahendra K

Numerade Educator

Problem 45

If the acceleration of a particle is the function of distance as $a=f(x)$. Then:
(a) the velocity must be the function of displacement
(b) the velocity versus displacement graph cannot be a straight line
(c) the velocity may be the function of displacement
(d) the acceleration versus displacement graph may be straight line

Mahendra K

Numerade Educator

Problem 46

A particle moves as such whose acceleration is given by $a=3 \sin 4 t$, then :
(a) the initial velocity of the particle must be zero
(b) the acceleration of the particle becomes zero after each interval of $\frac{\pi}{4}$ second
(c) the particle does not come at its initial position after some time
(d) the particle must move on a circular path

Mahendra K

Numerade Educator

Problem 47

A particle moves along a straight line such that its position $x$ at any time $t$ is $x=3 t^{2}-t^{3}$, where $x$ is in metre and $t$ in second, then:
(a) at $t=0$ acceleration is $6 \mathrm{~m} / \mathrm{s}^{2}$
(b) $x$ -f curve has maximum at $8 \mathrm{~m}$
(c) $x$ -f curve has maximum at $2 \mathrm{~s}$
(d) both (a) and (c) are correct

Mahendra K

Numerade Educator

Problem 48

The motion of a body falling from rest in a resisting medium is described by the equation $\frac{d v}{d t}=a-b v$, where $a$ and $b$ are constant. The velocity at any time $t$ is:
(a) $a\left(1-b^{2 t}\right)$
(b) $\frac{a}{b}\left(1-e^{-b t}\right)$
(c) $a b e^{-t}$
(d) $a b^{2}(1-t)$

Mahendra K

Numerade Educator

Problem 49

A rectangular box is sliding on a smooth inclined plane of inclination $\theta$. At $t=0$, the box starts to move on the inclined plane. A bolt starts to fall from point $A$. Find the time after which bolt strikes the bottom surface of the box:
(a) $\sqrt{\left(\frac{2 i}{8 \cos \alpha}\right)}$
(b) $\sqrt{\left(\frac{2 i}{g \sin \alpha}\right)}$
(c) $\sqrt{\left(\frac{2 i}{g}\right)}$
(d) $\sqrt{\left(\frac{l}{8}\right)}$.

Mahendra K

Numerade Educator

Problem 50

A point moves in a straight line under the retardation $k v^{2}$. If the initial velocity is $u$, the distance covered in $t$ second is :
(a) kut
(b) $\frac{1}{k} \log k u t$
(c) $\frac{1}{k} \log (1+k u t)$
(d) $k \log k u t$

Mahendra K

Numerade Educator

Problem 51

An object moves, starting from rest through a resistive medium such that its acceleration is related to velocity as $a=3-2 v$. Then :
(a) the terminal velocity is $1.5$ unit
(b) the terminal velocity is 3 unit
(c) the slope of $a-v$ graph is not constant
(d) initial acceleration is 2 unit

Mahendra K

Numerade Educator

Problem 52

A stone is released from a balloon moving upward with velocity $v_{0}$ at height $h$ at $t=0 .$ The speed-time graph is:

Mahendra K

Numerade Educator

Problem 53

If the velocity of a moving particle is $v \propto x^{n}$ where $x$ is the displacement, then:
(a) when $x=0$, the velocity and acceleration are zero
(b) $n>\frac{1}{2}$
(c) $n<\frac{1}{2}$
(d) (a) and
(b) are correct

Mahendra K

Numerade Educator

Problem 54

Which of the following statements is correct?
(a) When air resistance is negligible, the time of ascent is less than the time of descent
(b) When air resistance is not negligible, time of ascent is less than the time of descent
(c) When air resistance is not negligible, the time ascent is greater than the time of descent
(d) When air resistance is not negligible, the time of ascent is lesser than the time of descent

Mahendra K

Numerade Educator

Problem 55

A particle is projected veritically upward : n vacuum with a speed $40 \mathrm{~m} / \mathrm{s}$ then velocity of particle when it reaches at maximum height 2 s before, is : (Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )
(a) $20 \mathrm{~m} / \mathrm{s}$
(b) $4.2 \mathrm{~m} / \mathrm{s}$
(c) $9.8 \mathrm{~m} / \mathrm{s}$
(d) none of these

Mahendra K

Numerade Educator

Problem 56

A juggler keeps on moving four balls in the air throws the balls in regular interval of time. When one ball lea"es his hand (speed $=20 \mathrm{~m} / \mathrm{s}$ ), the position of other balls will be: (Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )
(a) $10 \mathrm{~m}, 20 \mathrm{~m}, 10 \mathrm{~m}$
(b) $15 \mathrm{~m}, 20 \mathrm{~m}, 15 \mathrm{~m}$
(c) $5 \mathrm{~m}, 15 \mathrm{~m}, 20 \mathrm{~m}$
(d) $5 \mathrm{~m}, 10 \mathrm{~m}, 20 \mathrm{~m}$

Mahendra K

Numerade Educator

Problem 57

Balls are thrown vertically upward in such a way that the next ball is thrown when the previous one is at the maximum height. If the maximum height is $5 \mathrm{~m}$, the number of balls thrown per minute will be:
(Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )
(a) 60
(b) 40
(c) 50
(d) 120

Mahendra K

Numerade Educator

Problem 58

A ball is dropped vertically from a height $d$ above the ground. It hits the ground and bounces up vertically to a height $d / 2 .$ Neglecting subsequent motion and air resistance, its speed $v$ varies with the height $h$ above the ground as :

Mahendra K

Numerade Educator

Problem 59

A ball is projected vertically upwards. If resistance due to air is ignored, then which of the following graphs represents the velocity-time graph of the ball during its flight?

Mahendra K

Numerade Educator

Problem 60

An object is thrown upward with a velocity $u$, then displacement-time graph is:

Mahendra K

Numerade Educator

Problem 61

A particle $P$ is sliding down a frictionless hemispherical bowl. It passes the point $A$ at $t=0 .$ At this instant of time, the horizontal component of its velocity is $v .$ A bead $Q$ of the same mass as $P$ is ejected from $A$ at $t=0$ along the horizontal string $A B$, with the speed $v .$ Friction between 66 the bead and the string may be neglected. Let $t_{p}$ and $t_{Q}$. be the respective times by $P$ and $Q$ to reach the point $B$. Then:
(a) $t_{P}<t_{Q}$
(b) $t_{p}=\lg$
(c) $t_{p}>t_{Q}$
(d) $\frac{t_{p}}{t_{Q}}=\frac{\text { length of arc } A C B}{\text { length of chord } A B}$

Mahendra K

Numerade Educator

Problem 62

Two stones $A$ and $B$ are dropped from a multistoried building with a time interval $t_{0}$, where $t_{0}$ is smaller than the time taken by $A$ to reach the floor. At $t=t_{0}$, stone $A$ is dropped. After striking the floor, stone comes to rest. The separation between stones plotted against the time lapse $t$ is best represented by:

Mahendra K

Numerade Educator

Problem 63

A balloon going upward with a velocity of $12 \mathrm{~m} / \mathrm{s}$ is at a height of $65 \mathrm{~m}$ from the earth surface at any instant. Exactly at this instant a packet drops from it. How much time will the packet take in reaching the surface of earth ? $\left(g=10 \mathrm{~m} / \mathrm{s}^{2}\right)$
(a) $7.5 \mathrm{sec}$
(b) $10 \mathrm{sec}$
(c) $5 \mathrm{sec}$
(d) none of these

Mahendra K

Numerade Educator

Problem 64

A stone is released from a balloon moving upward with velocity $v_{0}$ at height $h$ at $t=0 .$ Which of the following graphs is best representation of velocity-time graph for the motion of stone?

Mahendra K

Numerade Educator

Problem 65

A particle is projected at angle $60^{\circ}$ with the horizontal with speed $10 \mathrm{~m} / \mathrm{s}$ then equation of directrix is:
(Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )
(a) $y=5$
(b) $x=5$
(c) $x=10$
(d) $x+y=5$

Mahendra K

Numerade Educator

Problem 66

Three particles of equal masses are located at the vertices of an equilateral triangle whose side equals $a$. They all strart moving simultaneously with constant speed $v$ with the first point heading continuously for second, the second for third and third for first. Then:
(a) the distance travelled by each particle is $2 a / 3$
(b) at every instant before collision the momentum of the system is zero
(c) the force on each particle is perpendicular to velocity of the particle at any instant before collision
(d) all the above

Saman Zulfiqar

Numerade Educator

Problem 67

Eight particles are situated at the vertices of a regular octagon having edge length $10 \mathrm{~cm}$. They all start moving simultaneously with equal constant speed $1 \mathrm{~cm} / \mathrm{s}$ heading towards each other all the time. Then :
(a) momentum of system does not remain constant
(b) kinetic energy of the system remains constant after collision
(c) they will collide after time $\left(\frac{10 \sqrt{2}}{\sqrt{2}-1}\right)$ second
(d) every particle moves with constant acceleration

Mahendra K

Numerade Educator

Problem 68

A particle $P$ is at the origin starts with velocity $\overrightarrow{\mathrm{u}}^{7}=(2 \hat{i}-4 \hat{j}) \mathrm{m} / \mathrm{s}$ with constant acceleration
$\left(3 \hat{\mathrm{i}}+5^{\prime}\right) \mathrm{m} / \mathrm{s}^{2}$. After travelling for 2 second, its distance from the origin is :
(a) $10 \mathrm{~m}$
(b) $10.2 \mathrm{~m}$
(c) $9.8 \mathrm{~m}$
(d) $11.7 \mathrm{~m}$

Mahendra K

Numerade Educator

Problem 69

At an instant $t$, the co-ordinates of a particles are $x=a t^{2}, y=b t^{2}$ and $z=0 .$ The magnitude of velocity of particle at an instant $t$ is :
(a) $t \sqrt{a^{2}+b^{2}}$
(b) $\frac{v}{\sqrt{2}}$
(c) $\frac{V}{\sqrt{3}}$
(d) $2 t \sqrt{a^{2}+b^{2}}$

Mahendra K

Numerade Educator

Problem 70

If $x=a(\cos \theta+\theta \sin \theta)$ and $y=a(\sin \theta-\theta \cos \theta)$ and $\theta$
increases at uniform rate $\omega$. The velocity of particle is:
(a) $a \omega$
(b) $\frac{a^{2} \theta}{\omega}$
(c) $\frac{a \theta}{\omega}$
(d) $a \theta \omega$

Mahendra K

Numerade Educator

Problem 71

If co-ordinates of a moving point at time $t$ are given by $x=a(t+\sin t)$, and $y=a(1-\cos t)$, then :
(a) the slope of acceleration time graph is zero
(b) the slope of velocity-time graph is constant
(c) the direction of motion makes an angle $t / 2$ with $x$ -axis
(d) all the above

Saman Zulfiqar

Numerade Educator

Problem 72

A particle moves along the positive branch of the curve $y=\frac{x^{2}}{2}$ where $x=\frac{t^{2}}{2}$, where $x$ and $y$ are measured in metre and $t$ in second. At $t=2 \sec$, the velocity of the particle is:
(a) $(2 \hat{1}-4 \hat{j}) \mathrm{m} / \mathrm{sec}$
(b) $\left.\left(2 \hat{1}+4^{2}\right)\right) \mathrm{m} / \mathrm{sec}$
(c) $(2 \hat{1}+2 \hat{j}) \mathrm{m} / \mathrm{sec}$
(d) $(4 \hat{1}-2 \hat{j}) \mathrm{m} / \mathrm{sec}$

Mahendra K

Numerade Educator

Problem 73

The velocity of a particle moving in the $x$ - $y$ plane is given by
$$
\frac{d x}{d t}=8 \pi \sin 2 \pi t \quad \text { and } \quad \frac{d y}{d t}=5 \pi \cos 2 \pi t
$$
where $t=0, x=8$ and $y=0$. The path of the particle is:
(a) a straight line
(b) an ellipse
(c) a circle
(d) a parabola

Mahendra K

Numerade Educator

Problem 74

A light rigid rod is placed on a smooth horizontal surface. Initially the end $A$ begins to move vertically upward with constant velocity $v_{0}$ and centre of the rod
upward with a velocity $v_{0} / 2$ having downward acceleration $a_{0} / 2$, the other end moves downward with :
(a) zero initial velocity having zero acceleration
(b) zero initial velocity having $a_{0}$ downward acceleration
(c) non-zero initial velocity and zero acceleration
(d) none of the above

Mahendra K

Numerade Educator

Problem 75

At the top of the trajectory of a projectile, the directions of its velocity and acceleration are:
(a) parallel to each other
(b) inclined to each other at an angle of $45^{\circ}$
(c) anti parallel to each other
(d) perpendicular to each other

Mahendra K

Numerade Educator

Problem 76

A projectile is thrown at an angle of $\theta=45^{\circ}$ to the horizontal, reaches a maximum height of $16 \mathrm{~m}$. then :
(a) its velocity at the highest point is zero
(b) its range is $64 \mathrm{~m}$
(c) its range will decrease when it is thrown at an angle $30^{\circ}$
(d) (b) and (c) both are correct

Mahendra K

Numerade Educator

Problem 77

A heavy stone is thrown from a cliff of height $h$ in a given direction. The speed with which it hits the ground (air resistance may be neglected):
(a) must depend on the speed of projection
(b) must be larger than the speed of projection
(c) must be independent of the speed of projection
(d) (a) and (b) both are correct

Mahendra K

Numerade Educator

Problem 78

A particle is projected with speed $v$ at an angle $\theta$ $\left(0<\theta<\frac{\pi}{2}\right)$ above the horizontal from a height $H$ above the ground. If $v=$ speed with which particle hits the ground and $t=$ time taken by particle to reach ground, then:
(a) as $\theta$ increases, $v$ decreases and $t$ increases
(b) as $\theta$ increases, $v$ increases and $t$ increases
(c) as $\theta$ increases, $v$ remains same and $t$ increases
(d) as $\theta$ increases, $v$ remains same and $t$ decreases

Mahendra K

Numerade Educator

Problem 79

A particle of mass $m$ is projected with a velocity $v$ making an angle of $45^{\circ}$ with the horizontal. The magnitude of angular momentum of projectile about the point of projection when the particle is at its maximum height $h$ is :
(a) zero
(b) $\frac{m v h}{\sqrt{2}}$
(c) $\frac{m v h^{2}}{\sqrt{2}}$
(d) none of these

Mahendra K

Numerade Educator

Problem 80

Two particles are projected vertically upwards with the same velocity on two different planets with accelerations due to gravity $g_{1}$ and $g_{2}$ respectively. If they fall back to their initial points of projection after lapse of time $t_{1}$ and $t_{2}$ respectively, then :
(a) $t_{1} t_{2}=g_{1} g_{2}$
(b) $t_{1} g_{1}=t_{2} g_{2}$
(c) $\frac{t_{2} \sigma_{2}}{t_{2} g_{1}}-2$
(d) $t_{1}^{2}+t_{2}^{2}=g_{1}+g_{2}$

Mahendra K

Numerade Educator

Problem 81

A particle is projected from a horizontal plane to pass over two objects at heights $h$ and $k$ and a slant distance $d$ apart. The least possible speed of projection will be :
(a) $g(h+k+d)$
(b) $\sqrt{g(h+k+d)}$
(c) $h(g+k+d)$
(d) $\sqrt{h(g+h+d)}$

Saman Zulfiqar

Numerade Educator

Problem 82

The graph below shows one half period of a sinusoidal wave. It might represent the time dependence of :
(a) height of a projectile
(b) vertical component of a projectile's velocity
(c) $X$ -componeni of a projectile maving in uniform circular motion
(d) speed of an object subjectad to a force that grows linearly with time

Mahendra K

Numerade Educator

Problem 83

A number of particles are projected from a given point with equal velocities in different directions in the same vertical plane. At any instant, they will lie on:
(a) parabola
(b) circle
(c) hyperbola
(d) rectangle

Mahendra K

Numerade Educator

Problem 84

Two inclined planes are located as shown in figure. A particle is projected from the foot one frictionless plane along its line with a velocity just sufficient to carry it to top after which the particle slides down the other frictionless inclined plane. The total time it will take to reach the point $C$ is:
(a) $2 \mathrm{sec}$
(b) $3 \mathrm{sec}$
(c) $2 \sqrt{2} \mathrm{sec}$
(d) $4 \mathrm{sec}$

Mahendra K

Numerade Educator

Problem 85

Rain water is falling vertically downward with velocity
$v$. When velocity of wind is $u$ in horizontal direction, water is collected at the rate of $\mathrm{R} \mathrm{m}^{3} / \mathrm{s}$. When velocity of wind becomes $2 u$ in horizontal direction, the rate of collection of water in vessel is:
(a) $R$
(b) $\frac{R}{2}$
(c) $2 R$
(d) $\frac{R \sqrt{4 u^{2}+v^{2}}}{\sqrt{u^{2}+v^{2}}}$

Mahendra K

Numerade Educator

Problem 86

A particle is projected at an angle $\alpha$ with the horizontal from the foot of an inclined plane making an angle $\beta$ with horizontal. Which of the following expression holds good if the particle strikes the inclined plane normally?
(a) $\cot \beta=\tan (\alpha-\beta)$
(b) $\cot \beta=2 \tan (\alpha-\beta)$
(c) $\cot \alpha=\tan (\alpha-\beta)$
(d) $\cot \alpha=2 \tan (\alpha-\beta)$

Mahendra K

Numerade Educator

Problem 87

When the range of a projectile on an inclined plane is maximum then :
(a) the focus of the path is on the plane
(b) the focus of the path is below the plane
(c) the focus of the path is above the plane
(d) the focus of the path lies at any place

Mahendra K

Numerade Educator

Problem 88

If a number of particles are projected from the same point in the same plane so as to describe enual parabolas, then the vertices of their paths lie on a :
(a) parabola
(b) circle
(c) square
(d) rectangle

Mahendra K

Numerade Educator

Problem 89

The locus of foci of all parabolas described by the particles projected simultaneously from the same point with equal velocities but in different directions is a :
(a) circle
(b) parabola
(c) ellipse
(d) hyperbola

Mahendra K

Numerade Educator

Problem 90

A particle is projected at an angle $60^{\circ}$ with the horizontal with a speed $10 \mathrm{~m} / \mathrm{sec}$. Then latus
rectum is:
(Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )
(a) $5 \mathrm{~m}$
(b) $15 \mathrm{~m}$
(c) $10 \mathrm{~m}$
(d) 0

Mahendra K

Numerade Educator

Problem 91

A bus moves over a straight level road with a constant acceleration $a$. A boy in the bus drops a ball out side. The acceleration of the ball with respect to the bus and the earth are respectively:
(a) $a$ and $g$
(b) $a+g$ and $g-a$
(c) $\sqrt{a^{2}+g^{2}}$ and $g$
(d) $\sqrt{a^{2}+g^{2}}$ and $a$

Mahendra K

Numerade Educator

Problem 92

A man swims relative to water with a velocity greater than river flow velocity. Then:
(a) man may cross the river along shortest path
(b) man cannot cross the river
(c) man cannot cross the river without drifting
(d) none of the above

Mahendra K

Numerade Educator

Problem 93

Two cars move in the same direction along parallel roads. One of them is a $200 \mathrm{~m}$ long travelling with a velocity of $20 \mathrm{~m} / \mathrm{s}$. The second one is $800 \mathrm{~m}$ long travelling with a velocity of $7.5 \mathrm{~m} / \mathrm{s}$. How long will it take for the first car to overtake the second car?
(a) $20 \mathrm{~s}$
(b) $40 \mathrm{~s}$
(c) $60 \mathrm{~s}$
(d) $80 \mathrm{~s}$

Mahendra K

Numerade Educator

Problem 94

A motor boat covers the distance between two spots on the river banks in $t_{1}=8 \mathrm{~h}$ and $t_{2}=12 \mathrm{~h}$ in down stream and upstream respectively. The time required for the boat to cover this distance in still water will be :
(a) $6.9 \mathrm{hr}$
(b) $9.6 \mathrm{hr}$
(c) 69 second
(d) 96 second

Mahendra K

Numerade Educator

Problem 95

A man rows directly across a river in time $t$ second and rows an equal distance down the stream in $I$ second. The ratio of man's speed in still water to the speed of river water is :
(a) $\frac{t^{2}-T^{2}}{t^{2}+T^{2}}$
(b) $\frac{t^{2}+T^{2}}{t^{2}-T^{2}}$
(c) $\frac{T^{2}-t^{2}}{T^{2}+t^{2}}$
(d) $\frac{T^{2}+t^{2}}{T^{2}-t^{2}}$

Mahendra K

Numerade Educator

Problem 96

To a person going toward east in a car with a velocity of $25 \mathrm{~km} / \mathrm{hr}$, a train appears to move towards north with a velocity of $25 \sqrt{3} \mathrm{~km} / \mathrm{hr}$. The actual velocity of the train will be :
(a) $25 \mathrm{~km} / \mathrm{hr}$
(b) $50 \mathrm{~km} / \mathrm{hr}$
(c) $5 \mathrm{~km} / \mathrm{hr}$
(d) $53 \mathrm{~km} / \mathrm{hr}$

Mahendra K

Numerade Educator

Problem 97

A beautiful girl is going eastwards with a velocity of 4 $\mathrm{km} / \mathrm{hr}$. The wind appears to blow directly from the north. She doubles her speed and the wind seems to come from north east. The actual velocity of wind is :
(a) $4 \sqrt{2} \mathrm{~km} / \mathrm{hr}$ towards south east
(b) $4 \sqrt{2} \mathrm{~km} / \mathrm{hr}$ towards north west
(c) $2 \sqrt{2} \mathrm{~km} / \mathrm{hr}$ towards south east
(d) none of the above

Mahendra K

Numerade Educator

Problem 98

Rain drops fall vertically at a speed of $20 \mathrm{~m} / \mathrm{s}$. At what angle do they fall on the wind screen of a car moving with a velocity of $15 \mathrm{~m} / \mathrm{s}$ if the wind screen volocity inclined at an angle of $23^{\circ}$ to the vertical? $\left(\cot ^{-1} \frac{4}{3}=37^{\circ}\right)$
(a) $60^{\circ}$
(b) $30^{\circ}$
(c) $45^{\circ}$
(d) $90^{\circ}$

Mahendra K

Numerade Educator

Problem 99

A cyclist is moving with a constant acceleration of $1.2$ $\mathrm{m} / \mathrm{s}^{2}$ on a straight track. A racer is moving on a circular path of radius $150 \mathrm{~m}$ at constant speed of $15 \mathrm{~m} / \mathrm{s}$. Find the magnitude of velocity of racer which is measured by the cyclist has reached a speed of $20 \mathrm{~m} / \mathrm{s}$ for the position represented in the figure:
(a) $18.03 \mathrm{~m} / \mathrm{s}$
(b) $25 \mathrm{~m} / \mathrm{s}$
(c) $20 \mathrm{~m} / \mathrm{s}$
(d) $15 \mathrm{~m} / \mathrm{s}$

Mahendra K

Numerade Educator