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NCERT Exemplar Class 11 Physics Solutions Chapter 4 Motion in a Plane

NCERT Exemplar Class 11 Physics Solutions Chapter 4 Motion in a Plane

Edited By Safeer PP | Updated on Aug 08, 2022 05:20 PM IST

NCERT Exemplar Class 11 Physics solutions chapter 4 takes the concept of motion a step forward, by describing an object's motion in a two or three-dimensional plane. NCERT Exemplar Class 11 Physics chapter 4 solutions use the operations and graphical representation of vectors to help understand the motion in a plane. This chapter of NCERT Class 11 Physics Solutions revolves around a constantly accelerated motion in a plane and the relative velocity in two dimensions. Students would explore the concepts surrounding projectile motion and circular motion.

For a smooth learning curve and better understanding, students can avail the NCERT Exemplar Class 11 Physics solutions chapter 4 PDF Download function. These solutions are made by experts holding eminence in their respective fields. These solutions would help explain the concepts of 2D/3D motion well to score well in the final exams.

NCERT Exemplar Class 11 Physics Solutions Chapter 4 Motion in a Plane

NCERT Exemplar Class 11 Physics Solutions Chapter 4 MCQI

Question:4.1

The angle between A=\hat{i}+\hat{j} and B=\hat{i}-\hat{j} is
(a) 45^{o}
(b) 90^{o}
(c) -45^{o}
(d)180^{o}

Answer:

The answer is the option (b), 90^{o}
Explanation :
We know from the formula that A.B=\left \| A \right \|\left \| B \right \|\cos \theta
Given that \vec{A}=\hat{\iota }+\hat{J} and \vec{B}=\hat{\iota }-\hat{J}
\cos \theta =\frac{(\hat{\iota }+\hat{J})(\hat{\iota }-\hat{J})}{\sqrt{1^{2}+1^{2}}(\sqrt{1^{2}+(-1^{2})})}
=0=\cos 90
Hence the angle is 90^{o}

Question:4.2

Which one of the following statements is true?
(a) A scalar quantity is the one that is conserved in a process.
(b) A scalar quantity is the one that can never take negative values.
(c) A scalar quantity is the one that does not vary from one point to another in space.
(d) A scalar quantity has the same value for observers with different orientations of the axes.

Answer:

The answer is the option (d), A scalar quantity has the same value for observers with different orientations of the axes.
Explanation:
Unlike vector quantities, a scalar quantity has no dependence on the direction, and hence its value remains the same for all the orientations of the axes.

Question:4.3

Figure 4.1 shows the orientation of two vectors u and v in the XY plane.
capture-43
If u=a\hat{i}+b\hat{j} and
v=p\hat{i}+q\hat{j}
which of the following is correct?
(a) a and p are positive while b and q are negative.
(b) a, p and b are positive while q is negative.
(c) a, q and b are positive while p is negative.
(d) a, b, p and q are all positive.

Answer:

The answer is the option (b) a, p and b are positive while q is negative.
Explanation:
Now, \vec{u}=a\hat{\iota }+b\hat{J} and \vec{v}=p\hat{\iota }+q\hat{J}
For vector u, both the components a and b are in the positive direction of the x-axis. Hence, they are positive.
For vector v, p is in the positive direction, but q is in the opposite direction. Hence, p is positive, and q is negative.

Question:4.4

The component of a vector r along X-axis will have maximum value if
(a) r is along positive Y-axis
(b) r is along positive X-axis
(c) r makes an angle of 45° with the X-axis
(d) r is along negative Y-axis

Answer:

The answer is the option (b) r is along positive X-axis
Explanation: if a vector B forms an angle \theta along the x-axis. Then its components along the x-axis can be written as B Cos θ.
The maximum value of \cos \theta =1, which is when \theta =0.
Hence, for the vector r, the maximum value will be when it is along the positive x-axis.

Question:4.5

The horizontal range of a projectile fired at an angle of 15^{o} is 50 m. If it is fired with the same speed at an angle of 45^{o}, its range will be
a) 60 m
b) 71 m
c) 100 m
d) 141 m

Answer:

The answer is the option (c) 100 m
Explanation:
According to the formula:
R=\frac{u^{2}\sin {2\theta }}{g}
Given in the question:
\theta =15,R=50\; m
Putting in the formula we get:
u^{2}=100\; g
For \theta =45,
The value of the range is:
R=\frac{100\; g\times \sin{90}}{g}=100\; m

Question:4.6

Consider the quantities pressure, power, energy, impulse gravitational potential, electric charge, temperature, area. Out of these, the only vector quantities are
a) impulse, pressure, and area
b) impulse and area
c) area and gravitational potential
d) impulse and pressure

Answer:

The answer is the option (b) impulse and area
Explanation:
Impulse is a vector quantity and not a scalar quantity as the impulse is the rate of change of momentum, and it involves direction.
The area is a vector quantity

Question:4.7

In a two dimensional motion, instantaneous speed v0 is a positive constant. Then which of the following are necessarily true?
a) the average velocity is not zero at any time
b) average acceleration must always vanish
c) displacements in equal time intervals are equal
d) equal path lengths are traversed in equal intervals

Answer:

The answer is the option (d) equal path lengths are traversed in equal intervals.
Explanation: when instantaneous speed is positive and constant since speed is a scalar quantity equal paths will be covered in an equal amount of time.

Question:4.8

In a two dimensional motion, instantaneous speed v_{0} is a positive constant. Then which of the following are necessarily true?
a) the acceleration of the particle is zero
b) the acceleration of the particle is bounded
c) the acceleration of the particle is necessarily in the plane of motion
d) the particle must be undergoing a uniform circular motion

Answer:

The answer is the option (c) the acceleration of the particle is necessarily in the plane of motion
Explanation: In a two-dimensional motion, instantaneous speed is positive, and constant is given. Since velocity is constant, acceleration which is the rate of change in velocity is also constant. Hence, the acceleration will be in the plane of motion.

Question:4.9

Three vectors A,B and C add up to zero. Find which is false.
(a) (A * B) * C is not zero unless B,C are parallel
(b) (A * B).C is not zero unless B,C are parallel
(c) If A,B,C define a plane, (A * B) *C is in that plane
(d) (A^{*}B).C=\left | A \right |\left | B \right |\left | C \right |\rightarrow C^{2}=A^{2}+B^{2}

Answer:

The answer is the option (a) and (c)
Explanation:
Given \vec{A}+\vec{B}+\vec{C}=0
Option a : \vec{B}\times (\vec{A}+\vec{B}+\vec{C})=\vec{B}\times 0=0
0=\vec{B}\times \vec{A}+\vec{B}\times \vec{B}+\vec{B}\times \vec{C}
0=\vec{B}\times \vec{A}+0+\vec{B}\times \vec{C}
\vec{A}\times \vec{B}=\vec{B}\times \vec{C}(this cannot be zero)
Only if B and C are antiparallel or parallel \vec{B}\times \vec{C} will be zero
Hence for the whole quantity to be zero, \vec{B}\parallel \vec{C} should be true.
Option C : \vec{A}\times \vec{B}=\vec{X}
X is perpendicular to the planes which have vector A and vector B
Vector Y is perpendicular to the planes which have vector A and vector B
Vector Y is perpendicular to the plane containing X and C which is, in turn, the plane containing vectors A, B and C

Question:4.10

It is found that \left | A+B \right |=\left | A \right |. This necessarily implies,
(a)B=0
(b) A,B are antiparallel
(c) A,B are perpendicular
(d) A.B\leq 0

Answer:

The answer is the option (a) B=0
Explanation :
Squaring both sides and opening the brackets we get,
\left | \vec{A} \right |^{2}+\left |\overrightarrow{B} \right |^{2}+2\left | \vec{A} \right |\left | \overrightarrow{B} \right |\cos \theta =\left | \vec{A} \right |^{2}
\left | \overrightarrow{B} \right |^{2}+2\left | \vec{A} \right |\left | \overrightarrow{B} \right |\cos \theta =0
cos\theta=\frac{-B}{2A}
if B=0
cos\theta=0

NCERT Exemplar Class 11 Physics Solutions Chapter 4 MCQII

Question:4.11

Two particles are projected in air with speed v_{0}, at angles \theta _{1} and \theta _{2} to the horizontal, respectively. If the height reached by the first particle is greater than that of the second, then tick the right choices
a) angle of project: q_{1} > q_{2}
b) time of flight: T_{1} > T_{2}
c) horizontal range: R_{1} > R_{2}
d) total energy: U_{1} > U_{2}

Answer:

The answer is the option (a) and (b)
Explanation:
According to formula. Max height of a projectile is
H = \frac{u^{2} \sin^{2} {\theta }}{2g}
Option a :H1>H2
\sin ^{2}\theta 1>\sin ^{2}\theta 2
(\sin \theta 1+\sin \theta 2)(\sin \theta 1-\sin \theta 2)>0
So, (\sin \theta 1+\sin \theta 2)>0 \; or(\sin \theta 1-\sin \theta 2)>0
So, \theta 1>\theta 2 and \theta lies etween 0 and 90 degree i.e. acute
Option b :
\frac{T1}{T2}=\frac{\sin \theta 1}{\sin \theta 2}
T1\sin \theta 2=T2\sin \theta 1
Since \sin \theta 1>\sin \theta 2
T1>T2

Question:4.12

A particle slides down a frictionless parabolic (y=x^{2}) track (A - B - C) starting from rest at point A (Fig. 4.2). Point B is at the vertex of parabola and point C is at a height less than that of point A. After C, the particle moves freely in air as a projectile. If the particle reaches highest point at P, then
capture-412
(a) KE at P = KE at B
(b) height at P = height at A
(c) total energy at P = total energy at A
(d) time of travel from A to B = time of travel from B to P.

Answer:

The answer is the option (c)
Explanation: total energy at a point in the path remains the same as there is no friction between air and plane on paths A, B and C
As we know through the energy conservation law, total energy at point P will be same as total energy at point A.
Hence, option c is correct.

Question:4.13

Following are four different relations about displacement, velocity, and acceleration for the motion of a particle in general. Choose the incorrect one(s) :
(a) V_{av}=\frac{1}{2}\left [ V(t_{1})+V(t_{2}) \right ]
(b) V_{av}=\frac{r(t_{2})-r(t_{1})}{t_{2}-t_{1}}
(c) r=\frac{1}{2}\left ( V(t_{2})-V(t_{1}) \right )\left ( t_{2}-t_{1} \right )
(d) a_{av}=\frac{V(t_{2})-V(t_{1})}{t_{2}-t_{1}}

Answer:

The answer is the option (a) and (c)
Explanation:
Option a:
The given relation is correct when the acceleration is uniform
Option c:
\vec{r}=\frac{1}{2}\left ( \vec{v}(t_{2})-\vec{v}(t_{1}) \right )\div \left ( t_{2}-t_{1} \right )
This is the relationship given in the question, but it is not possible as the LHS and RHS dimensions \left [ M^{0}L^{1}T^{0} \right ] and \left [ M^{0}L^{1}T^{2} \right ]do not match and hence the relationship cannot be considered valid.

Question:4.14

For a particle performing uniform circular motion, choose the correct statement(s) from the following:
(a) Magnitude of particle velocity (speed) remains constant.
(b) Particle velocity remains directed perpendicular to radius vector.
(c) Direction of acceleration keeps changing as particle moves.
(d) Angular momentum is constant in magnitude but direction keeps changing.

Answer:

The answer is the option (a), (b), and (c)
Explanation:
Option a: speed is constant at all times in the case of uniform circular motion
Option b: in case of velocity, in a circular motion it is measured tangentially to the direction of motion of the particle which is in turn perpendicular to the radius.
Option c: the direction of the acceleration is in the direction of force always. This can concur with the newton’s second law of motion. So as the particle moves in the circular motion, the direction of force keeps on changing and hence that of acceleration also changes.

Question:4.15

For two vectors A and B, |A + B| = |A - B| is always true when
(a) \left | A \right |=\left | B \right |\neq 0
(b) \left | A \right |\perp \left | B \right |
(c) \left | A \right |= \left | B \right |\neq 0 and A and B are parallel or anti parallel
(d) when either \left | A \right | or \left | B \right | is zero.

Answer:

The answer is the option (b) and (d)
Explanation:
Given : \left | \vec{A}+\vec{B} \right |=\left | \vec{A}-\vec{B} \right |
Now we square both sides and open the brackets to get:
\left | \vec{A} \right |^{2}+\left | \vec{B} \right |^{2}+2\left | \vec{A} \right |\left | \vec{B} \right |\cos \theta =\left | \vec{A} \right |^{2}+\left | \vec{B} \right |^{2}-2\left | \vec{A} \right |\left | \vec{B} \right |\cos \theta
4\left | \vec{A} \right |\left | \vec{B} \right |\cos \theta=0
So, we have, \theta =90 degree and \left | \vec{A} \right |=\left | \vec{B} \right |=0
Hence option b and option d are correct.

NCERT Exemplar Class 11 Physics Solutions Chapter 4 Very Short Answer

Question:4.16

A cyclist starts from centre O of a circular park of radius 1km and moves along the path OPRQO as shown Fig. 4.3. If he maintains constant speed of 10 ms–1, what is his acceleration at point R in magnitude and direction?
capture-416

Answer:

To calculate acceleration in a circular motion, we have a formula.
A=\frac{v^{2}}{R}
Given, R=1000 \; m, v = 10\; m/s
Hence, A=10\times \frac{10}{1000}=\frac{1}{10}=0.1\; m/s^{2}

Question:4.17

A particle is projected in air at some angle to the horizontal, moves along parabola as shown in the figure, where x and y indicate horizontal and vertical directions respectively. Show in the diagram, direction of velocity and acceleration at points A, B, and C.
capture-417

Answer:

A particle travelling in a projectile motion will definitely have a path which is parabolic in nature. The velocity of such a particle is always measured tangential to the path of motion. Point B is where the particle reaches maximum height during the projectile motion.
capture-417-1
Here we have the vertical and horizontal components of B which are V_{y}=0 and V_{x}=u\cos \theta

Question:4.18

A ball is thrown from a roof top at an angle of 45^{o} above the horizontal. It hits the ground a few seconds later. At what point during its motion, does the ball have
a) greatest speed
b) smallest speed
c) greatest acceleration?

Answer:

capture-418
The ball is thrown from point O. The angle with the horizontal is 45 degree. As the height increases from O to A, the speed decreases. At B, the speed becomes equal to its initial speed. The height decreases further from B to C and becomes maximum at point C.
V_{x}=u\; \cos 45=\frac{u}{\sqrt{2}}\; m/s
a) greatest speed of the ball is at c as V_{y} and V_{x}=u\; \cos\; 45=\frac{u}{\sqrt{2}}\; m/s
b) slowest speed will be at point A. here V_{y}=0, and the horizontal speed is V_{x}=\frac{u}{\sqrt{2}}
c) the greatest acceleration will be the one caused by the gravitational force downward, which will be equal to g.

Question:4.19

A football is kicked into the air vertically upwards. What is its
a) acceleration
b) velocity at the highest point

Answer:

a) when the football is kicked into the air, the only force working on it is the gravitational force. Hence the acceleration will be caused due to this force which will be in the downward direction towards the centre of the Earth.
b) when the ball is thrown upward, no horizontal forces act on it and hence the horizontal component velocity becomes nil. The vertical component of velocity at the highest point is V_{y}=0. Hence, the net velocity when the ball reaches the highest point is also zero.

Question:4.20

A, B, and C are three non-collinear, non co-planar vectors. What can you say about direction of A(B\times C)?

Answer:

Given: A, B, and C are non-collinear and no co-planar vectors.
For finding out the direction, we used the right-hand grip rule.
For (\vec{B}\times\vec{C}), the direction will be in a plane perpendicular to the plane containing \vec{B} and \vec{C}.
However, for the direction of \vec{A}\times(\vec{B}\times\vec{C}): it will be perpendicular to both \vec{A} and the plane which has (\vec{B}\times\vec{C}).

NCERT Exemplar Class 11 Physics Solutions Chapter 4 Short Answer

Question:4.21

A boy travelling in an open car moving on a labelled road with constant speed tosses a ball vertically up in the air and catches it back. Sketch the motion of the ball as observed by a boy standing on the footpath. Give explanation to support your diagram.

Answer:

Let v be the vertical velocity, and u be the horizontal velocity of the car.
capture-421
In the case of the ball, as it has both horizontal and vertical components of velocity, it will have a parabolic path when a person standing on a footpath sees the trajectory.
capture-421-1
When a boy sitting in the same car sees the ball, the path will vertical up and down. However, he will be able to catch up with the ball provided the car moves at a constant velocity.

Question:4.22

A boy throws a ball in air at 60^{o} to the horizontal along a road with a speed of 10 m/s. Another boy sitting in a passing by car observes the ball. Sketch the motion of the ball as observed by the bot in the car, if car has a speed of 18 km/h. Give explanation to support your diagram.

Answer:

Given : u=36\; km/h=10\; m/s
capture-422
u_{x}=u\; \cos 60=5\; m/s
now, in the direction of the ball, the speed of the car is 18\times \frac{5}{18}=5\; m/s
since the horizontal speeds are the same, in the situation of a ball thrown by the boy as soon as the car passes him, they also cover equal horizontal distances.
capture-422-1
Now, when vertical components are considered, u_{y}=u\; \cos 30=5\sqrt{3}\; m/s and the motion of the ball is vertically up-down as seen by the boy.

Question:4.23

In dealing with motion of projectile in air, we ignore effect of air resistance on motion. This gives trajectory as a parabola as you have studied. What would the trajectory look like if air resistance is included? Sketch such a trajectory and explain why you have drawn it that way.

Answer:



stik1
When we are dealing with projectile motion generally we neglect the air resistance. But if air resistance is included the horizontal component of velocity will not be constant and obviously trajectory will change.
Due to air resistance, particle energy, as well as horizontal component of velocity, keep on decreasing making the fall steeper than rise as shown in the figure.
When we are neglecting air resistance path was symmetric parabola (OAC). When air resistance is considered path is asymmetric parabola (OAB).

Question:4.23

In dealing with motion of projectile in air, we ignore effect of air resistance on motion. This gives trajectory as a parabola as you have studied. What would the trajectory look like if air resistance is included? Sketch such a trajectory and explain why you have drawn it that way.

Answer:

capture-423
The horizontal and vertical components of the velocity decrease due to air resistance. So, as a result, the max height also becomes lesser than in the ideal case.
Now, R=u^{2}\sin 2\; \theta/g and max height H=u^{2}\sin^{2} \; \theta/2g
So, h1<h2\;and\; R1<R2
But, in the second case when h1<h2, due to smaller time taken to rise, the overall time of flight for both cases stands equal

Question:4.24

A fighter plane is flying horizontally at an altitude of 1.5 km with speed 720 km/h. At what angle of sight when the target is seen, should the pilot drop the bomb in order to attack the target?

Answer:

capture-424
u=720\; km/h=200\; m/s
we assume that the bomb is dropped by the pilot t seconds vertically above Q before the target T
vertical component velocity of the bomb will be zero and the horizontal component value will be equal to that of the plane. So, the bomb covers distance TQ as a free fall.
Now, u=0,H=1.5,g=10
So, H=ut+\frac{1}{2}gt^{2}=0+\frac{1}{2}10\times t\times t=1500
So, we get, t=10\sqrt{3}\; s
ut is the distance covered by plane or the bomb equal to PQ
hence, PQ=ut=200\sqrt{3}\; m
now, \tan \theta =\frac{TQ}{PQ}=\sqrt{\frac{3}{4}}=23^{o}42'

Question:4.26

Given below in column I are the relations between vectors a, b, and c and in column II are the orientations of a, b, and c in the XY plane. Match the relation in column I to correct orientations in column II.

Column IColumn II
a) a+b=ccapture-426-1
5b) a-c=bcapture-426-ii
c) b - a = ccapture-426-iii

d) a+b+c=0capture-426-iv

Answer:

a) \vec{a}+\vec{b}=\vec{c} matches with (iv)
b) \vec{c}+\vec{b}=\vec{a} matches with (iii)
c) \vec{a}+\vec{c}=\vec{b} matches with (i)
d) \vec{a}+\vec{c}+\vec{b}=0 matches with (ii)

Question:4.27

If \left | A \right |=2 and \left | B \right |=4, then match the relations in column I with the angle \theta between A and B in column II.
Column I Column II
(a) A.B=0 (i) \theta =0
(b) A.B=+8 (ii) \theta =90^{o}
(c) A.B=4 (iii) \theta =180^{o}
(d) A.B=-8 (iv) \theta =60^{o}

Answer:

a) Matches with (ii)
\left | \vec{A} \right |\left | \vec{B} \right |\; \cos \theta =0
\cos \theta =0
\theta =90
b) Matches with (i)
\left | \vec{A} \right |\left | \vec{B} \right |\; \cos \theta =8
2\times4 \cos \theta =8
\cos \theta =1
\theta =0
c) Matches with (iv)
\left | \vec{A} \right |\left | \vec{B} \right |\; \cos \theta =4
2\times4 \cos \theta =84
\cos \theta =\frac{1}{2}
\theta = 60
d) Matches with (iii)
\left | \vec{A} \right |\left | \vec{B} \right |\; \cos \theta =-8
2\times4 \cos \theta =-8
\cos \theta =-1
\theta =180

Question:4.28

If \left | A \right |=2 and \left | B \right |=4 then match the relations in column I with the angle \theta between A and B in column II.
Column I Column II

(a)\left | A\times B \right |=0 (i) \theta =30^{o}
(b) \left | A\times B \right |=8 (ii) \theta =45^{o}
(c) \left | A\times B \right |=4 (iii) \theta =90^{o}
(d) \left | A\times B \right |=4\sqrt{2} (iv) \theta =0^{o}

Answer:

Given : \left | \vec{A} \right |=2 and \left | \vec{B} \right |=4
a) Matches with (iv)
\left | \vec{A}\times \vec{B} \right |=0
\left | \vec{A} \right |\left | \vec{B} \right |\sin \theta =0
2\times4\; \sin \theta =0
\sin \theta =\sin 0
\theta =0
b) matches with (iii)
\left | \vec{A}\times \vec{B} \right |=8
\left | \vec{A} \right |\left | \vec{B} \right |\sin \theta =8
2\times4\; \sin \theta =8
\sin \theta =1
\theta =90
c) matches with (i)
\left | \vec{A}\times \vec{B} \right |=4
\left | \vec{A} \right |\left | \vec{B} \right |\sin \theta =4
2\times4\; \sin \theta =4
\sin \theta =\frac{1}{2}
\theta =30
d) matches with (ii)
\left | \vec{A}\times \vec{B} \right |=4\sqrt{2}
\left | \vec{A} \right |\left | \vec{B} \right |\sin \theta =4\sqrt{2}
2\times4\; \sin \theta =4\sqrt{2}
\sin \theta =\frac{1}{\sqrt{2}}
\theta =45

NCERT Exemplar Class 11 Physics Solutions Chapter 4 Long Answer

Question:4.29

A hill is 500 m high. Supplies are to be sent across the hill using a canon that can hurl packets at a speed of 125 m/s over the hill. The canon is located at a distance of 800 m from the foot of hill and can be moved on the ground at a speed of 2 m/s so that its distance from the hill can be adjusted. What is the shortest time in which a packet can reach on the ground across the hill? Take g = 10 m/s2.

Answer:

capture-429
Packet speed = 125 m/s, height of hill = 500m
In order to cross the hill, the vertical component of the packet should be reduced to make the height of 500m attainable. The distance between canon and hill should also be half of that of the packet’s range.
V^{2}-u^{2}=2gh
U_{y}=\sqrt{2gh}=\sqrt{10000}=100\; m/s
Now, U=u{_{x}}^{2}+u{_{y}}^{2}
u{_{x}}^{2}=125\times125-(100\times100)
u_{x}=75\; m/s
now we consider the packet's vertical motion,
v_{y}=u_{y}+gt
t= total time of flight = 10 sec
so, v_{y}=75\times10=750
so the distance between canon and hill is 750m
distance for which the canon needs to move=800-750=50m
time taken for the canon to move 50m=\frac{50}{2}=25 sec
so the total time taken by the packet =25+10+10=45\; seconds

Question:4.3

A gun can fire shells with maximum speed v_{0} and the maximum horizontal range that can be achieved is R=v{_{0}}^{2}/g
capture-430
If a target farther away by \Delta x beyond R has to be hit with the same gun as shown in the figure. Show that it could be achieved by rating the gun to a height at least
h=\Delta x[1+\frac{\Delta x}{R}]

Answer:

capture-430-1
The solution to this problem can be given in two ways:
i) the target is present at the horizontal distance of (R+\Delta x) and is h meters below the projection point. Y=-h
ii) the motion of projectile starting from point P till reaching point T. vertical height covered is -h and horizontal range is \Delta x
max range of a projectile is given by R=\frac{v^{2}}{g}
here, \theta =45
we assume that the gun is raised to a height h to hit the target T
total range =(R+\Delta x)
the horizontal component of velocity is, v\; \cos \theta
horizontal velocity at P=V_{x}=-V\; \cos \theta
vertical velocity at P=V_{y}=V\sin \theta
now, h=ut+\frac{1}{2}at^{2}
So, h=-V\; \sin \theta t+\frac{1}{2}gt^{2}\; \; \; \; \; \; ---------------(1)
V\; \cos \theta .t=(R+\Delta x)
t=\frac{(R+\Delta x)}{V\; \cos \theta }
Substituting the value of t in (1) we get,
h=-V\; \sin \theta \left ( \frac{(R+\Delta x)}{V\; \cos \theta } \right )+\frac{1}{2}g\left ( \frac{(R+\Delta x)}{V\; \cos \theta } \right )^{2}
h=-(R+\Delta x)+\frac{1}{R}(R^{2}+\Delta x^{2}+2R\; \Delta x)=-R-\Delta x+R+\frac{\Delta x^{2}}{R}+2\Delta x
h=\Delta x[1+\frac{\Delta x}{R}]
hence proved.

Question:4.31

A particle is projected in air at an angle \beta to a surface which itself is inclined at an angle \alpha to the horizontal.
a) find an expression of range on the plane surface
b) time of flight
c) \beta at which range will be maximum

Answer:

1622170825079


a) expression of range on the plane surface
Now, a_{y}=-g\; \cos \alpha and a_{x}= g\; \sin \alpha
Y=0 at O and P,
So
U_{y}=v\; \sin \beta where t = T
We calculate Time of Flight Part (b) before Part (a)
b) motion of particle along Y axis
s=ut+\frac{1}{2}gt^{2}
s=0,u=v\; \sin \beta , g=a=-g\; \cos \alpha ,t=T
0=v\; \sin \beta .T+\frac{1}{2}(-g\; \cos \alpha )T^{2}
T[v\; \sin \beta -T.\frac{g}{2}\cos \alpha ]=0
\frac{Tg}{2}.\cos \alpha =v\sin \beta
So, T=2v\; \sin \beta /g.\cos \alpha
a) Now we continue the part a
T=2v\; \sin \beta /g.\cos \alpha
s=ut+\frac{1}{2}gt^{2}
L=v\; \cos \beta (T)+\frac{1}{2}(-g\; \sin \alpha )T^{2}
L=2\; v^{2}\; \sin \beta[\cos \beta \cos \alpha -\sin \beta \sin \alpha ]/g \cos ^{2}\alpha
L=2\; v^{2}\; \sin \beta \times \cos [\alpha +\beta ]/g \cos ^{2}\alpha
c) on the axis X, L will be maximum when \sin \beta \cos [\alpha +\beta ] will be maximum
let z=\sin \beta \cos [\alpha +\beta ]
=\sin \beta [\cos \beta \cos \alpha -\sin \beta \sin \alpha ]
=\frac{1}{2}[\cos \alpha\; 2\sin\; \beta +\sin \alpha \cos \; 2\; \beta-\sin \alpha ]
=\frac{1}{2}[\sin(2\beta +\alpha )-\sin\; \alpha ]
In order to make Z maximum, we put [\sin(2\beta +\alpha )-\sin\; \alpha ]=1
Opening the brackets, we get (2\beta +\alpha )=90
2\beta =90-\alpha
\beta =45-\frac{\alpha }{2}radian

Question:4.32

A particle falling vertically from a height hits a plane surface inclined to horizontal at an angle \theta with speed v_{o} and rebounds elastically (Fig 4.7). Find the distance along the plane where it will hit second time.
capture-432

Answer:

The particle rebounds from P. when it strikes plane inclined at v0 speed. Hence the speed of particle after it rebounds from P will be v0 We assume the new axis X’OX and YOY’ axis at P as origin ‘O’. The components of g and v0 in the new OX and OY axis are:
capture-432-1
Focusing on the motion of the particle from O to A,
s=ut+\frac{1}{2}gt^{2}
Here, t=T which is the time of flight
0=T[v\; \cos\; \theta -\frac{1}{2}g\; \sin\; \theta \; T]
So, either T=0\; or \left [ v\; \cos\; \theta -\frac{1}{2}g\; \sin\; \theta \; T \right ]=0
S_{x}=u_{x}t+\frac{1}{2}a_{x}t^{2}
L=[\frac{2v}{g}]v\; \sin \theta +\frac{1}{2}g\; \sin\; \theta [\frac{2v}{g}]^{2}
L=\frac{4v^{2}}{g}.\sin \theta

Question:4.33

A girl riding a bicycle with a speed of 5 m/s towards north direction, observes rain falling vertically down. If she increases her speed to 10 m/s, rain appears to meet her at 45^{o} to the vertical. What is the speed of the rain? In what direction does rain fall as observed by a ground based observer?

Answer:

Let the north direction be i and south direction be j
The velocity of rain is v=a\hat{i}+b\hat{J}
Case 1 (v=5i)
capture-49950
The velocity of rain with respect to the girl is:
v_{r}-v_{g}=(a\hat{i}+b\hat{j})-5\hat{i}=(a-5)\hat{i}+b\hat{j}
Since the horizontal component is zero, a-5=0 \; or\; a=5
Case 2 (v=10i)
capture-49950-1
v_{r}-v_{g}=(a\hat{i}+b\hat{j})-10\hat{i}=(a-10)\hat{i}+b\hat{j}
The angle of rain appears to be 45 degrees.
\tan 45=\frac{b}{a}=\frac{b}{-5}
b=-5
So,
\left | v_{r} \right |=\sqrt{5^{2}+(-5)^{2}}=\sqrt{50}=5\sqrt{2}\; m/s

Question:4.34

A river is flowing due east with a speed 3m/s. A swimmer can swim in still water at a speed of 4 m/s (Fig. 4.8).
capture-434
(a) If swimmer starts swimming due north, what will be his resultant velocity (magnitude and direction)?
(b) If he wants to start from point A on south bank and reach opposite point B on north bank,
(a) which direction should he swim?
(b) what will be his resultant speed?
(c) From two different cases as mentioned in (a) and (b) above, in which case will he reach opposite bank in shorter time?

Answer:

a) If swimmer starts swimming due north, what will be his resultant velocity
capture-434-1
V_{s}=4\; m/s due north
V_{r}=4\; m/s due east
Now since both the directions are perpendicular,
\left | V_{r} \right |^{2}=4^{2}+3^{2}=5\; m/s
\tan \; \theta =\frac{V_{r}}{V_{s}}=0.75=36^{o}54' in the North direction

b) The swimmer wants to start from point A on the south bank and reaches the opposite point B on the north bank

capture-434-2
The swimmer makes an angle \theta with the north.
From the figure we have the relation,
V^{2}=v{_{s}}^{2}-v{_{r}}^{2}=16-9=7
Henecev=\sqrt{7}\; m/s
Now we calculate the value of θ through the below formula,
\tan \theta =\frac{v_{r}}{v}=\frac{3\sqrt{7}}{7}=1.13
So, \theta =48^{o}29'30'' in th edirection from North to West
c) we need to find from the above two scenarios that for the swimmer to reach the opposite bank in the shorter time
we know that the velocity component perpendicular to the river is 4m/s
let us assume the width of the river to be ‘w’
Time taken - North
\frac{w}{4}=t1
time taken in part b) when v=\sqrt{7}\; m/s
\frac{w}{\sqrt{7}}=t2
taking ratio,
\frac{t1}{t2}=\frac{(\frac{w}{4})}{(\frac{w}{\sqrt{7}})}
4\; t1=\sqrt{7}\; t2
Now as, 4>\sqrt{7}
t1<t2
So, the swimmer will take a shorter time in case a)

Question:4.35

A cricket fielder can throw the cricket ball with a speed vo. If he throws the ball while running with speed u at an angle \theta to the horizontal, find
a) the effective angle to the horizontal at which the ball is projected in air as seen by a spectator
b) what will be time of flight?
c) what is the distance from the point of projection at which the ball will land?
d) find \theta at which he should throw the ball that would maximise the horizontal range as found c)
e) how does \theta for maximum range change if u>v_{o}=v_{o} and u<v_{o}
f) how does \theta in e) compare with that for u = 0?

Answer:

a) u is the horizontal velocity with which the cricketer runs. The ball is thrown by him while running and hence the speed of ball also contains a component of the cricketer’s speed.
capture-435
U_{x}=u+v\; \cos \; \theta
Vertical component,
U_{y}=v\; \sin \; \theta
\tan \theta =\frac{v\; \sin \theta }{u+v\cos \theta }
\theta =\tan ^{-1}[\frac{v\; \sin\; \theta }{u+v\; \cos \; \theta }]
b) Time of flight
S_{y}=U_{y}t+\frac{1}{2}a_{y}t^{2}
Since the ball returns back to the same position, Sy = 0
U_{y}=v\; \sin \theta
So,0=v\; \sin \theta (T)-\frac{1}{2}g\; T^{2}
T[v\; \sin \theta -\frac{1}{2}g\; T]=0
Since T cannot be zero, we have
T=2v\; \sin \frac{\theta }{g}
c) Maximum range
for the max range, the condition is
\frac{dR}{d\theta} =0
d\frac{\left \{ \frac{v}{g}[2u]\sin \; \theta +\frac{v}{g}\; \sin 2\theta \right \}}{d\; \theta }=0
\theta =\cos ^{-1}\left [ \frac{-u\pm \sqrt{u^{2}+8\; v_{0}^{2}}}{4v_{0}} \right ]
\cos \; \theta =\left [ \frac{-u\pm \sqrt{u^{2}+8\; v_{0}^{2}}}{4v_{0}} \right ]
e) In the case when u=v,\cos \theta =
\frac{-v_{0}\pm \sqrt{v_{0}^{2}+8v_{0}^{2}}}{4v_{0}}
=\frac{-v_{0}+3v_{0}}{4v_{0}}
\cos\; \theta =-1+\frac{(-3)}{4}
\cos\; \theta =\frac{1}{2} (as \theta is taken as an acute angle here)
hence, \theta =60^{o}
for the case of u << v
\cos \theta =-u+\frac{(-2\sqrt{2}v)}{4v}
Since \theta is an acute angle here,
as u << v here, we can neglect the last term
\cos \theta =\frac{1}{\sqrt{2}}
\theta =\frac{\pi }{4}
For u >> v,
\cos\; \theta =\frac{-u+(-u)}{4v}
\cos\; \theta =0=\cos 90
\theta =\frac{\pi }{2}
f) when u=0
\cos\; \theta =\frac{-u\pm \sqrt{u^{2}+8v_{0}^{2}}}{4v_{0}}
\cos\; \theta =\frac{2\sqrt{2}v}{4v}=\frac{1}{\sqrt{2}}
\cos\; \theta =45
\cos\; \theta =\frac{\pi }{4}

Question:4.36

Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in Cartesian co-ordinates A=A_{x}\hat{i}+A_{y}\hat{j} where \hat{i} and \hat{j} are unit vector long x and y directions, respectively and A_{x} and A_{y} are corresponding components of A (fig.4.9). Motion can also be studied by expressing vectors in circular polar co-ordinates as A=A_{r}\hat{r}+A_{\theta }\hat{\theta } where \hat{r}=\frac{r}{r}=\cos \theta \hat{i}+\sin \theta \hat{j} and \hat{\theta }=-\sin \theta \hat{i}+\cos \theta \hat{j} are unit vectors along direction in which 'r' and '\theta ' are increasing.
capture-436
(a) Express \hat{i} and \hat{j} in terms of \hat{r} and \hat{\theta }
(b) Show that both \hat{r} and \hat{\theta } are unit vectors and are perpendicular to each other.
(c) Show that \frac{d}{dt}(\hat{r})=\omega \hat{\theta } where \omega =\frac{d\theta }{dt} and \frac{d}{dt}(\theta )=-\omega \hat{r}
(d) For particle moving along a spiral given by r=a\theta \hat{r}, where a=1 (unit), find dimensions of 'a'.
(e) Find velocity and acceleration in polar vector represention for particle moving along spiral described in (d) above.

Answer:

a) \hat{r}=\cos\; \theta \hat{i}+\sin\; \theta \hat{J}\; \; \; \; \; \; -------(1)
\hat{\theta }=-\sin\; \theta \hat{i}+\cos \theta \hat{J}\; \; \; \; ----------(2)
When we multiply 1 by \sin \theta and 2 by \cos \theta, we get :
\hat{r}\sin \theta +\hat{\theta }\cos \theta =(\sin^{2}\theta +\cos^{2}\theta )\hat{J}
\hat{r}\sin \theta +\hat{\theta }\cos \theta =\hat{J}\; \; \; \; \; \; \; \; ---------(3)
When we multiply 1 by \cos \theta and 2 by \sin \theta we get :
\hat{r}\cos \theta =\cos^{2}\theta \hat{i}+\sin \theta \cos \theta \hat{J}\; \; \; \; \; -----(4)
\hat{\theta }\sin \theta =-\sin^{2}\theta \hat{i}+\sin \theta \cos \theta \hat{J}\; \; \; \; \; -----(5)
Subtracting equation 5 from 4 and comparing the coefficients we get,
\hat{r}\cos\theta -\hat{\theta }\sin \theta =\hat{i}
\hat{r}\sin\theta -\hat{\theta }\cos \theta =\hat{J}
b) from equation 1 and 2, through the dot product method we get,
\hat{r}.\hat{\theta }=\left ( \cos\; \theta \hat{i}+\sin \theta \hat{J} \right ).\left ( -\sin\; \theta \hat{i}+\cos\; \theta \hat{J} \right )
\left | \hat{r} \right |\left | \hat{\theta } \right |\cos\; \theta =0
Since LHS elements cannot be zero,
\cos \theta =0
and \theta =90
c) \hat{r}=\cos\; \theta \hat{i}+\sin \theta \hat{J}
\frac{d\; \hat{r}}{dt}=\frac{d(\cos\; \theta \hat{i}+\sin \theta \hat{J})}{dt}
=\left ( -\sin \theta \hat{i} +\cos \theta \hat{J}\right )\frac{d\; \theta }{dt}
Since, \omega =\frac{d\theta }{dt}
\frac{d\hat{r}}{dt}=\omega \hat{\theta }
d) \hat{r}=\left | \hat{a} \right |\left | \hat{\theta } \right |\hat{r}
now looking at the dimensions of the quantities on the LHS and the RHS,
[a]=\frac{[\hat{r}]}{[\hat{\theta }][\hat{r}]}
=\frac{[M^{0}L^{1}T^{0}]}{[M^{0}L^{0}T^{0}][M^{0}L^{0}T^{0}]}
=[M^{0}L^{1}T^{0}]
e) a=1
\hat{r}=\hat{\theta }[\cos\; \theta \hat{i}+\sin\; \theta \hat{J}]
V=\frac{d\hat{r}}{dt}=\frac{d\theta }{dt}\hat{r}+\theta (-\sin \theta \hat{i}+\cos \theta \hat{J})\frac{d\; \theta }{dt}
V=\omega \hat{r}+\theta .\hat{\theta }\omega
\vec{a}=\frac{dv}{dt}
=\frac{d(\omega \hat{r}+\theta .\hat{\theta }\omega )}{dt}
=d^{2}\frac{\theta }{dt^{2}}\hat{r}+d\frac{\theta }{dt}d\; \hat{r}\frac{\hat{r}}{dt}+d^{2}\frac{\theta }{dt^{2}}(\theta .\hat{\theta })+d\frac{\theta }{dt}(\theta .\hat{\theta })
=d^{2}\frac{\theta }{dt^{2}}\; \hat{r}+\omega \left ( \frac{-\sin\; \theta \hat{i}\; d}{dt}+\frac{\cos\; \theta \hat{J}\; d\; \theta }{dt} \right )+\frac{d^{2}\theta }{dt^{2}}(\theta .\hat{\theta })+\frac{\omega \; d\; \theta }{dt}(\theta .\hat{\theta })
=\frac{d^{2}\theta }{dt^{2}\hat{r}}+\omega ^{2}\hat{\theta }+\frac{d^{2}\theta }{dt^{2}}(\theta .\hat{\theta })+\omega ^{2}\hat{\theta }+\omega ^{2}(-\hat{r})
\vec{a}=\left ( \frac{d^{2}\theta }{dt^{2}-\omega ^{2}} \right )\hat{r}+\left ( \frac{2\omega ^{2+}d^{2}\theta }{dt^{2}\theta } \right )\hat{\theta }

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Main Subtopics in NCERT Exemplar Class 11 Physics Solutions Chapter 4 Motion in a Plane

Class 11 Physics NCERT Exemplar Solutions Chapter 4 Includes The Following Topics:

  • 4.1 Introduction
  • 4.2 Scalars and vectors
  • 4.3 Multiplication of vectors by real numbers
  • 4.4 Addition and subtraction of vectors — graphical method
  • 4.5 Resolution of vectors
  • 4.6 Vector addition — analytical method
  • 4.7 Motion in a plane
  • 4.8 Motion in a plane with constant acceleration
  • 4.9 Relative velocity in two dimensions
  • 4.10 Projectile motion
  • 4.11 Uniform circular motion

What Will The Students Learn in NCERT Exemplar Class 11 Physics Solutions Chapter 4?

  • Students would be able to link the concepts of motion in the previous chapter to this chapter and broaden their learning scope.
  • This chapter will also introduce students to the vector operations used to determine the motion of an object with specific dimensions.
  • NCERT Exemplar Class 11 Physics chapter 4 solutions would provide a graphical and analytical representation of vectors for a better understanding of motion.
  • It answers questions and queries related to relative velocity in dimensions and constant acceleration.
  • NCERT exemplar Class 11 Physics solutions chapter 4 also offers the opportunity to learn about the broad topic of projectile motion to determine a ball's trajectory in a cricket match or the launching of a satellite and its path.

NCERT Exemplar Class 11 Physics Solutions Chapter-Wise

Important Topics To Cover For Exams in NCERT Exemplar Solutions for Class 11 Physics Chapter 4

· NCERT Exemplar Class 11 Physics solutions chapter 4 highlights the previous chapter of motion in a straight line to explain many of motion's concepts and terms in a plane. It talks about the different arithmetic expressions, including graphical and theoretical methods to add, subtract, multiply, and resolve vectors.

· This chapter covers dimensions' characteristics and how they're used to explain objects' straightforward motion in a 2D or 3D plane, be it in a constantly accelerated motion or a uniform circular motion.

· NCERT Exemplar Class 11 Physics solutions chapter 4 also talks about the broad applications of projectile motion in determining the future position or the direction of an object in motion and the significance of uniform circular motion. NCERT Exemplar Class 11 Physics chapter 3 solutions provides a comprehensive learning perspective that extends the relative velocity concepts.

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2. How can these solutions help in final exams?

NCERT Exemplar Class 11 Physics Solutions Chapter 4 can help you use the applications of motions in real life, understand and determine the position of objects in projectile motion, and help give an excellent academic performance in your final exams.

3. What are the essential topics of this chapter?

This chapter's essential topics are the Scalars and vectors, operations on vectors, motion in a plane, relative velocity, projectile, and uniform circular motion.

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

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2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

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K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

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33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

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67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

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be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

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Weight fraction of solute

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Fraction of solute present in water

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Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

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half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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