Practice Problem in Physics for the JEE Main and Advanced

Abhay Kumar

Chapter 3

Motions in Two and Three Dimensions - all with Video Answers

Educators

Section 1

Section A

Problem 1

A particle moves along positive branch of the curve $y=x / 2$; where $x=t^{3} / 3, x$ and $y$ are measured in metres and $t$ in seconds, then
(a) the velocity of particle at $t=1 \mathrm{~s}$ is $\hat{i}+(1 / 2) \hat{j}$
(b) the velocity of particle at $t=1 \mathrm{~s}$ is $(1 / 2) \hat{i} \mid \hat{j}$
(c) the acecleration of particle at $t=2 \mathrm{~s}$ is $2 \hat{i}+\hat{j}$
(d) the acceleration of particle at $t=2 \mathrm{~s}$ is $\hat{i}$ ? $2 \hat{j}$

Hast Aggarwal

Hast Aggarwal

Numerade Educator

Problem 2

A particle moves in such a way that its positon vector at any time $t$ is $\vec{r}-t \hat{i}\left|\frac{1}{2} t^{2} \hat{j}\right| \hat{k}$. Find as a function of time (a) the velocity, (b) the speed, (c) the acecleration, (d) the magnitudc of the accelcration, (c) the magnitude of the component of acceleration along velocity (called tangential acecleration), (I) the magnitude of the component of acceleration perpendicular to velocity (called normal acecleration or centripctal accelcration).

Hast Aggarwal

Numerade Educator

Problem 3

(A) A partielc is moving in three dimensions. Its positon vector is given by $\vec{r}-6 \hat{i} \mid(3 \mid 4 t) \hat{j} \quad\left(3 \mid 2 t, t^{2}\right) \hat{k}$
Distance are in metres, and the time, $t$, in scconds.
(a) What is the velocity vector at $t=+3 \mathrm{sec}$ ?
(b) What is the specd (in m/s) at $t=+3$ scc?
(c) What is the acceleration vector and what is its magnitude (in $\mathrm{m} / \mathrm{s}^{2}$ ) at $t=+3$ scc?
(B) Now, the particle is moving only along the $z$ -axis, and its positon is given by, $\left(\begin{array}{lll}t^{2} & 2 t & 3) \hat{k}\end{array}\right.$
at what time does the particle stand still?

Hast Aggarwal

Numerade Educator

Problem 4

The radius vector of a point $A$ relative to the origin varies with time $t$ as $\vec{r}-a t \hat{i}-b t^{2} \hat{j} ;$ where $a$ and $b$ arc positive constants. Find:
(a) equation of the point's trajectory $y(x)$,
(b) lime dependence of velocity $\vec{v}$ and acceleration $\bar{f}$ and their moduli,
(c) lime dependence of the angle $\theta$ bctween $\bar{v}$ and $\vec{f}$, and
(d) mean velocity vector averaged over the first $t$ scconds of motion and its modulus.

Hast Aggarwal

Numerade Educator

Problem 5

A point moves according to the law $x=a t, y=a t(1-\alpha t)$ where $a$ and $\alpha$ are positive constants and $t$ is time Find the moment al which angle between velocity voctor and acecleration vector is $\frac{\pi}{4}$.

Hast Aggarwal

Numerade Educator

Problem 6

A particle is at origin at $t=0 .$ Calculate the distance from origin at time $3 \pi / 2 \omega$.

Nidhi Singhi

Numerade Educator

Problem 7

A particle starts from origin at $t=0$ and moves in the $x-y$ planc with a constant acccleration $\alpha$ in the $y$ -dircction. The equation of motion is $y=k x^{2}$. Find its velocity component along $x$ -dircction.

Hast Aggarwal

Numerade Educator

Problem 8

A particle moves in $x y$ plane with a velocity given by $\vec{v}=(8 t-2) \hat{i}+2 \hat{j} .$ If it passes through the point $(14,4)$ at $t=2 \mathrm{scc}$, then find its cquation of the path.

Hast Aggarwal

Numerade Educator

Problem 9

A particle moves in the $x y$ planc as $\vec{v}=a \hat{i} \mid b x \hat{j}$, wherc $\hat{i}$ and $\hat{j}$ are the unit vectors along $x$ and $y$ axis. The particle starts from origin at $t=0$. Find the radius of curvature of the particle as a function of $x$.
(a) $\frac{a^{2} \mid b^{2} x^{2}}{b a}$
(b) $\frac{a}{b}\left[1+\left(\frac{b x}{a}\right)^{2}\right]^{3 / 2}$
(c) $\frac{b}{a}\left[1+\left(\frac{a x}{b}\right)^{2}\right]^{3 / 2}$
(d) Nonc

Hast Aggarwal

Numerade Educator

Problem 10

If the velocity of a particle is defined by $\vec{r}-(2 t+1) \hat{i}+3 \hat{j}$ and its position vector at $t=\mathrm{ls}$ is $\vec{r}-4 \hat{i}+3 \hat{j}$, determine the path of the particle.

Hast Aggarwal

Numerade Educator

Problem 11

A particle moves according to the equation $y=a x-b x^{2}$. Only gravitation field is present. Find the initial velocity.

Hast Aggarwal

Numerade Educator

Problem 12

A particle moves in the $x y$ plane with constant acceleration $\vec{a}$, directed towards the negative $y$ -axis. 'Ihe equation of motion of the particle has the form $y=b x-c x^{2}$, where $b$ and $c$ are positive constants. Find the velocity of the particle at the origin of coordinates.

Hast Aggarwal

Numerade Educator

Problem 13

The height $y$ and the distance $x$ along the horizontal planc of a projectile on a certain planct (assuming flat surfacc) with no surrounding atmosphere are given by $x=6 t$ and $y=8 t-5 t^{2}$, wherc $x$ and $v$ are in metre and time $t$ is in second. Find the velocity with which the body is projected, the maximum height attained and the range of the projectile. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$

Hast Aggarwal

Numerade Educator

Problem 14

Shots are fired simultaneously from the top and bottom of a vertical cliff at angles $\alpha$, and $\beta$ and they strike an object simultaneously at the same point. Show that if the horizontal distance of the object from the cliff is $a$, the height of the cliff is
(a) $\frac{a(\cot \alpha-\cot \beta)}{\cot \alpha \cot \beta}$
(b) $a(\tan \beta-\tan \alpha)$
(c) $\frac{a \tan \alpha}{\tan \beta}$
(d) $a(\cot \alpha-\cot \beta)$

Hast Aggarwal

Numerade Educator

Problem 15

A stonc is thrown from a point at a distance $a$ from a wall of hcight $b$. If it just clears the wall, the maximum height reached by the stone, if the angle of projection is $\alpha$, is
(a) $\frac{a^{2} \sec ^{2} \alpha}{4(a \sec \alpha \quad b)}$
(b) $\frac{a^{2} \tan ^{2} \alpha}{4(a \tan \alpha}$
(c) $\frac{a^{2} \tan \alpha}{4(a b \cot \alpha)}$
(d) $\frac{a^{2} \tan \alpha}{4}$

Hast Aggarwal

Numerade Educator

Problem 16

A stone is projected so as to pass two walls of heights $a$ and $b$ at distances $b$ and $a$, respectively from the point of projection. Show that the angle of projection must be greater than $\tan ^{-1}(3)$. Find the range.

Hast Aggarwal

Numerade Educator

Problem 17

A gun is aimed at a monkey on a troc. Immediately, the monkcy falls down frecly. Docs the bullet hit the monkey?

Hast Aggarwal

Numerade Educator

Problem 18

A particle is projected from the origin in such a way that it passes through a given point $P(a, b)$. What is the minimum required speed to do so?

Nidhi Singhi

Numerade Educator

Problem 19

'Two particles $A$ and $B$ are projected from the same point in different directions in such a manner that vertical components of their initial velocities are same :
(a) Find ratio of time of flight
(b) Find ratio of range.

Hast Aggarwal

Numerade Educator

Problem 20

A ball is projected horizontally from a height of $100 \mathrm{~m}$ from the ground with a speed of $20 \mathrm{~m} / \mathrm{s}$. Find:
(a) the time taken to reach the ground,
(b) the horizontal distance it covers before striking the ground, and
(c) the velocity with which it strikes the ground. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$.

Hast Aggarwal

Numerade Educator

Problem 21

Two particles are projected simultancously with the samc speed $v$ in the same vertical planc with angles of clevation $\theta$ and $2 \theta$, wherc $\theta<45^{\circ}$. At what time will their velocities be parallel?

Hast Aggarwal

Numerade Educator

Problem 22

A body is thrown horizontally with a velocity $\sqrt{2 g h}$ from the top of a tower of height $h$. It strikes the level ground through the foot of the tower at a distance $x$ from the tower. Find the value of $x$.

Hast Aggarwal

Numerade Educator

Problem 23

Two stones are projected so as to reach the same distance from the point of projcetion on a horizontal surface. The maximum height reached by one cxecods the other by an amount equal to half the sum of the heights reached by them. What arc the angles of projection?

Hast Aggarwal

Numerade Educator

Problem 24

A particle is projected to pass over a tower of height $h$. The produet of the two possible times taken to pass over the tower is
(a) $\frac{2 u}{g}$
(b) $\frac{2 h}{g}$
(c) $\frac{2 u}{g}\left(\sin \alpha_{1}+\sin \alpha_{2}\right)$
(d) Nonc

Hast Aggarwal

Numerade Educator

Problem 25

The velocity of a projectile when it is at the greatest height is $\sqrt{\frac{2}{5}}$ times its velocity when it is at hal of its greatest height. Determine its angle of projection.

Hast Aggarwal

Numerade Educator

Problem 26

A stone is projected at an angle $\alpha$ to the horizontal from the top of a tower of height $3 h$. If the stone reaches a maximum height $h$, above the tower, show that it reaches the ground at a distance $6 h$ cot $\alpha$ from the foot of the tower.

Hast Aggarwal

Numerade Educator

Problem 27

If the angles of the elevation of a particle moving in aparabola are $\alpha$ and $\beta$ from the point of projection and the point where it reaches the horizontal plane from the point of projection, show that $\tan \alpha+\tan \beta=\tan \theta ;$ where $\theta$ is the angle of projection.

Nidhi Singhi

Numerade Educator

Problem 28

The horizontal range of a bullct, when fircd at $45^{\circ}$, is $1200 \mathrm{~m}$. If it is fired from a lorry, moving at $24 \mathrm{~km} / \mathrm{h}$ with the same velocity and clevation, then find its range. (Take $\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^{2}$ )

Nidhi Singhi

Numerade Educator

Problem 29

From a point at a height $h$ above the oround, a particle $A$ is projected with a velocity $v$ in an upward direction making an angle $\theta$ with the horizontal. $\Lambda$ nother particle $B$ is projected from the same point with the same velocity $v$ but, in a direction directly opposite to $A$. Show that the two particles hit the ground al a distanec $\frac{2 v}{g} \cos \theta \sqrt{\left(v^{2} \sin ^{2} \theta+2 g h\right)}$ apart.

Nidhi Singhi

Numerade Educator

Problem 30

A particle moves along a given trajectory with a tangential acceleration $a_{t}=a_{0} \cos \alpha$ where $\alpha$ is the angle between the direction of motion and the $x$ -axis. Find how the velocity of the particle depends on $x$ provided its velocity is zero at the point $x=0$.

Nidhi Singhi

Numerade Educator

Problem 31

A jet plane flics horizontally at a hcight $h$ at a speed $v .$ An antiaircraft gun fires a shell at the planc when it is vertically above the gun. Show that the minimum mursle speed required to hit the plane is $\sqrt{v^{2} 12 g h}$ at an angle $\tan ^{-1} \frac{\sqrt{2 g h}}{v}$.

Nidhi Singhi

Numerade Educator

Problem 32

A particle is to be projected so as to just pass through three equal rings of diameter $d$ and placed in parallel vertical planes at distances $a$ apart with their highest points at a height $h$ above the point of projection. Prove that the clcvation of projection is tan $\frac{\sqrt{h d}}{a}$.

Nidhi Singhi

Numerade Educator

Problem 33

A projectile is fired from the base of a conc-shaped hill. The projectile grazos the vertex and strikes the hill again at the base. If $\alpha$ be the half-angle of the conc, $h$ its height, $u$ the initial velocity of projection and $\theta$ angle of projection, show that tan $\theta=2 \cot \alpha$ and $u^{2}-g h\left(\frac{1}{2} \tan ^{2} \alpha+2\right)$

Nidhi Singhi

Numerade Educator

Problem 34

Three particles are projected simultaneously and in the same vertical plane with different velocities and at different angles. Show that the area of the triangle formed by the particles at time $t$ is proportional to $t^{2}$.

Nidhi Singhi

Numerade Educator

Problem 35

Two particles move in a gravitational field with constant acceleration $g \cdot \Lambda$ the initial moment, the particles were located at one point and moved with velicities $v_{1}=3 \mathrm{~m} / \mathrm{s}$ and $v_{2}=4 \mathrm{~m} / \mathrm{s}$ horizontally in oposite direction. Find the distance between the particles when their velocity vectors are mutually perpendicular: $\left(\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^{2}\right)$

Nidhi Singhi

Numerade Educator

Problem 36

A particle when projected in vertical planc moves along the fixed smooth surfacc with initial velocity $20 \mathrm{~m} / \mathrm{s}$ at an angle of $60^{\circ}$, so that its normal reaction on the surlace remains isro throughout the motion. Then find the slope of the tangent to the surface at height $5 \mathrm{~m}$ from the point of projoction $A$.

Nidhi Singhi

Numerade Educator

Problem 37

The speed of a particle travelling in circle of radius $r$ varies as $v=2 s^{2}$ where $s$ is the distance travelled. Calculate the magnitude and direction of acceleration after one revolution.

Nidhi Singhi

Numerade Educator

Problem 38

A particle is moving in a circle such that its kinetic energy varies as $K=t^{3}$. Calculate the direction and magnitude of acceleration after $1 \mathrm{sec}$. if the radius of the circle is equal to $r$

Nidhi Singhi

Numerade Educator

Problem 39

A particle is moving in a circle of radius $R . \Lambda \mathrm{t} t=0$ its speed is $v_{0}$ and during its motion speed
varies as $\frac{d v}{d t}-\frac{-k v^{2}}{R}$ where $k$ is a positive constant. Find the speed of the particle: (a) after $1 \mathrm{sec}$; (b) after 1 revolution.

Nidhi Singhi

Numerade Educator

Problem 40

A particle is moving with a constant angular acccleration $\alpha=4 \mathrm{rad} / \mathrm{scc}^{2}$ in a circular path. At timc $t=0$ the particle was at rest. Find the time at which the magnitudes of centripctal acceleration and tangential acccleration are cqual

Nidhi Singhi

Numerade Educator

Problem 41

In a circular motion of a particle, the tangential acceleration of the particle is given by $a_{t}=2 t \mathrm{~m} /$ $\mathrm{sec}^{2}$, where $t$ is the time of motion. The radius of the circle described is $R=2 \mathrm{~m}$. 'lhe particle is initially at rest. $\Lambda$ t time $t$, the total acceleration of the particle makes an angle $45^{\circ}$ with radial acceleration. Choose the correct alternatives.
(a) 'lhe value of $t$ is $2^{3 / 3} \mathrm{~s}$.
(b) The value of $t$ is $2^{1 / 3} s$.
(c) The total acceleration of the particle when $t=1$ s is $\sqrt{17} / 2 \mathrm{~m} / \mathrm{s}^{2}$.
(d) The total acceleration of the particle when $t=1$ s is $1 / 4 \mathrm{~m} / \mathrm{s}^{2}$.

Nidhi Singhi

Numerade Educator

Problem 42

A particle is projected with vel. $u$ from the ground making an angle $\theta$ with horizontal find the radius of curvature: (a) at the point of projection, (b) at point of strike, (c) at maximum height
(d) at the point where the velocity of projectile makes an angle $\theta / 2$ with horizontal

Nidhi Singhi

Numerade Educator

Problem 43

A bus and a car start from rest and run along a straight linc with constant acccleration of $2 \mathrm{~m} / \mathrm{s}^{2}$ and $3 \mathrm{~m} / \mathrm{s}^{2}$ respectively, with the car behind the bus at distance of $100 \mathrm{~m}$. How long will it take beforc both the bus and car travel side by sidc?

Nidhi Singhi

Numerade Educator

Problem 44

Two trains $A$ and $B, 125 \mathrm{~m}$ and $100 \mathrm{~m}$ long are moving in opposite dircctions on parallcl tracks. The velocity of train $B$ is threc times that of train $A$. The trains take 4 sec to pass cach other calculate the velocity of cach train.

Nidhi Singhi

Numerade Educator

Problem 45

At the sight of grecn signal a car starts with a uniform acceleration of $4 \mathrm{~m} / \mathrm{s}^{2}$. At the same instant a bus travelling with a constant velocity $30 \mathrm{~m} / \mathrm{sec}$ overtakes and passes the car.
(i) At what distance from the starting point will the car overtake the bus?
(ii) How fast will the car $B$ travelling at that instant?

Nidhi Singhi

Numerade Educator

Problem 46

$N$ particles $A, B, C, D, E, \ldots \ldots$ are situated at the corners of $N$ sided regular polygon of side $L$. Each of the particles moves with constant speed $v \cdot A$ always has its velocity along $A B, B$ along $B C$ and $C$ along $C D$ and so on. $\Lambda$ t what time and where will the particles meet each other?

Nidhi Singhi

Numerade Educator

Problem 47

$P$ is a point moving with constant speed $10 \mathrm{~m} / \mathrm{s}$ such that its velocity vector always maintains an angle $60^{\circ}$ with line $O P$ as shown in figure $(O$ is a fixed point in space). The initial distance between $O$ and $P$ is $100 \mathrm{~m} . \Lambda$ fter what time shall $p$ reach $O ?$

Nidhi Singhi

Numerade Educator

Problem 48

Swimmer's problem or River-boat problem:
I.cl $V_{1 v}=$ velocity of water w.r.t ground Vs, $w=$ velocity of swimmer w.r.t. water $V_{s}=$ velocity of swimmer w.r.l. ground
Now, $A C=V_{s}^{\prime} \times$ crossing time $t$
$\Rightarrow \quad(A B+B C)=\left(V_{s, w}^{\prime}+V_{w}^{\prime}\right)$
$\Rightarrow \quad(d \hat{j}+x \hat{i})=\left[\left(-V_{s, w}^{\prime} \sin \theta\right) \hat{i}+\left(V_{s, 18} \cos \theta\right) \hat{j}+i_{w} \hat{i}\right] t$
$\left.\Rightarrow \quad(x \hat{i}+d \hat{j})=\left(V_{v}-V_{s, 1 \mathrm{~F}} \sin \theta\right) t \hat{i}+\left(\ddot_{t, v} \cos \theta\right) t \hat{j}\right)$
i.c. $\quad x=\left(V_{w}-V_{A, v} \sin \theta\right) t$
and $d=\left(V_{s, v} \cos \theta\right) t \quad \Rightarrow \quad i-\frac{d}{\left(\sigma_{n, w}{\cos \theta)}\right.} \quad \ldots(3)$

Nidhi Singhi

Numerade Educator

Problem 49

A man can row a boat with $4 \mathrm{~km} / \mathrm{hr}$. in water if he is crossing a river. The current is $2 \mathrm{~km} / \mathrm{h}$.
(a) In what direction will his boat be headed, if he wants to reach a point on the other bank directly opposite to starting point.
(b) If width of the river be $4 \mathrm{~km}$ how long will the man take to cross the river with the condition in (a).
(c) In what direction should he head the boat if he wants to cross the river in shortest time and what is this minimum timc?
(d) How long will it take him to row $2 \mathrm{~km}$ up stream and then back to his starting point?

Nidhi Singhi

Numerade Educator

Problem 50

A swimmer wants to reach point $B$ on the opposite bank of a river flowing at a speed as shown in figure. What minimum speed relative to water should the swimmer have so that he can reach point $B^{\prime}$ ? In which direction should he swim?

Nidhi Singhi

Numerade Educator

Problem 51

Boatman wants to reach from point $A$ to point $B$ as shown in figure. $\Lambda t$ what angle $\theta$ should he head his boat. Given: $\left|\vec{v}_{b r}\right|-4 \mathrm{~m} / \mathrm{s}=$ magnitude of velocity of boatman with respect to river $\vec{v}_{r}-2 \mathrm{~m} / \mathrm{s}$ in the dircetion shown.

Nidhi Singhi

Numerade Educator