Class 11th Chapter -4 Motion in a Plane |Physics| NCERT Solution | Edugrown

NCERT Exercises

Question 1.
State, for each of the following physical quantities, if it is a scalar or a vector: volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency,displacement, angular velocity.
Answer:
Scalars : volume, mass, speed, density, number of moles, angular frequency.
Vectors: Acceleration, velocity, displacement, angular velocity.

Question 2.
Pick out the two scalar quantities in the following lists : force, angular momentum, work, current, linear momentum, electric field,average velocity, magnetic moment, relative velocity.
Answer:
If Work and current are the scalar quantities in the given list.

Question 3.
Pick out the only vector quantity in the following
list : Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.
Answer:
Since, Impulse = change in momentum = force x time. As momentum and force are vector quantities, hence impulse is a vector quantity, hence impulse is a vector quantity.

Question 4.
State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful:
(a) adding any two scalars.
(b) adding a scalar to a vector of the same dimensions
(c) multiplying any vector by any scalar.
(d) multiplying any two scalar.
(e) adding any two vectors.
(f) adding a component of a vector to the same vector.
Answer:
(a) No, adding any two scalars is not meaningful because only the scalars of same dimensions (i.e. of same nature) can be added.

(b) No, adding a scalar to a vector of the same dimension is not meaningful because a scalar cannot be added to a vector.

(c) Yes, multiplying any vector by any scalar is meaningful algebraic operation. It is because when any vector is multiplied by any scalar, then we get a vector having magnitude equal to scalar number times the magnitude of the given vector, g. when acceleration a is multiplied by mass m, we get force F = ma which is a meaningful operation.

(d) Yes, the product of two scalars gives a meaningful result g. when power P is multiplied by time t, then we get work done (W) i.e. W = Pt, which is a meaningful algebraic operation.

(e) No, as the two vectors of same dimensions (i.e. of the same nature) can only be added, so addition of any two vectors is not a meaningful algebraic operation.

(f) No, a component of a vector can be added to the same vector only by using the law of vector addition. So, the addition of a vector to the same vector is not a meaningful operation.

Question 5.
Read each statement below carefully and state with reasons, if it is true or false:
(a) The magnitude of a vector is always a scalar.
(b) Each component of a vector is always a scalar.
(c) The total path length is always equal to the magnitude of the displacement vector of a particle,
(d) The average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time,
(e) Three vectors not lying in a plane can never add up to give a null vector.”
Answer:
(a) True; because magnitude of a vector is a pure number.
(b) False; as each component of a given vector is always a vector.
(c) True; only if the particle moves along a straight line in the same direction otherwise false.
(d) True; because the total path length is either greater than or equal to the magnitude of the displacement vector, so the average speed is greater or equal to the magnitude of average velocity.
(e) True; as they cannot be represented by the three sides of a triangle taken in the same order. Here the resultant of any two vectors will be in the plane of these two vectors only and it cannot balance the third vector which is in a different plane. Two vectors can cancel each other’s effect only if they are equal in magnitude and opposite in direction.

Question 6.
Establish the following vector inequalities geometrically or otherwise:
(a) |a + b|<|d| + |b|
(b) |o + b|>||d| + |b||
(c) |d-b|<|a| + |b|
(d) |a-b|>||d|-|b||
When does the equality sign above apply?
Answer:
Consider that the two vectors 5 and b are represented by OP and OQ. The addition of the two vectors e. a +b is given by OR as shown in figure (i).

Question 7.
Given  ++  + = 0, which of the following statements are correct
(a)  ++  and  must each be a null vector,
(b) The magnitude of () equals the magnitude of ( +  ),
(c) The magnitude of  can never be greater than the sum of the magnitudes of ,  and ,
(d) +  must lie in the plane of  and  , if  and  are not collinear, and in the line of  and , if they are collinear?
Answer:


Question 8.
Three girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in figure. What is the magnitude of the displacement vector for each? For which girl is this equal to the actual length of path skate?

Answer:
Let the three girls be A, B and C. Let PAQ, PBQ and PCQ be the paths followed by A, B and C respectively. Radius of the circular track = 200 m.
As all the girls start from point P and reach at
.’. Displacement vector for each girl = 
So the magnitude of the displacement vector for each girls =| |
Diameter of the circular ice ground
= 2 x 200 = 400 m.
From figure, it is clear that for girl B, the magnitude of the displacement vector is equal to the actual length of the path skated.

Question 9.
A cyclist starts from the center O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the center along QO as shown in figure. If the round trip takes 10 min, what is the
(a) net displacement,
(b) average velocity, and
(c) average speed of the cyclist

Answer:
(a) Net displacement is zero as both initial and final positions are same.

Question 10.
On an open ground, a motorist follows a track that turns to his left by an angle of 60° after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
Answer:
Suppose that the motorist starts from the point O along the initial direction OX. After covering OA = 500 m, he turns to his left through 60° along AL and takes the first turn at the point A. After travelling a distance AB = 500 m along AL, he turns to his left through 60° and takes the second turn at the point B

(1) At the third turn : The displacement of the motorist at the third turn is OC. From the points A and B, draw AN₁ and BN₂ perpendiculars to OC. Then,

(2) At the sixth turn : Since at the sixth turn, the motorist reaches the starting point, the displacement of the motorist is a null vector e. if S₂ is path length upto the sixth turn, then S₂ = 6 x 500 = 3,000m.

(3) At the eighth turn : At the eighth turn, the displacement of the motorist will be OB. From the point A, draw AN₃ perpendicular to . Then,


Question 11.
A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is
(a) the average speed of the taxi,
(b) the magnitude of average velocity? Are the two equal?
Answer:
Magnitude of the displacement = 10 km
Distance covered = 23 km
Time taken = 28 min

Clearly, the average speed and the magnitude of average velocity are not equal. They are equal only for a straight path.

Question 12.
Rain is falling vertically with a speed of 30 m s^-1. A woman rides a bicycle with a speed of 10 m s^-1 in the north to south direction. What is the direction in which she should hold her umbrella?
Answer:
In figure, the rain is falling along OA with speed 30 m s^-1 and woman rider is moving along OS with speed 10 m s^-1 i.e. OA = 30 m s^-1 & OB = 10 m s^-1. The woman  rider can protect herself from the rain if she holds her umbrella in the direction of relative velocity of rain w.r.t. woman. To do so apply equal and opposite velocity of woman on the rain i.e. impress the velocity 10 m s^-1 due North on rain which is represented by OC.

Question 13.
A man can swim with a speed of 4.0 km h^-1 in still water. How long does he take to cross a river 1.0 km wide if the river flows steadily at km h^_1 and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other bank?
Answer:
Speed of man, υ_x = 4 km h^-1
Distance travelled = 1 km
Speed of river = 3 km h^-1

Question 14.
In a harbour, wind is blowing at the speed of 72 km h^-1 and the flag on the mast of a boat anchored in the harbour flutters along the N-E direction. If the boat starts moving at a speed of 51 km h^-1 to the north, what is the direction of the flag on the mast of the boat?
Answer:
When the boat is anchored in the harbour, the flag flutters along the N-E direction. It shows that the velocity of wind is along the north-east direction. When the boat starts moving, the flag will flutter along the direction of relative velocity of wind w.r.t. boat. Let _wb be the relative velocity of wind w.r.t. boat and β be the angle between _wb and _w. Refer Fig.

Question 15.
The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball thrown with a speed of 40 m s^-1 can go without hitting the ceiling of the hall?
Answer:

Question 16.
A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high above the ground can the cricketer throw the same ball?
Answer:

Question 17.
A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone make 14 revolutions in 25 s, what is the magnitude and direction of the acceleration of the stone?
Answer:

The direction of centripetal acceleration is along the string directed towards the center of circular path.

Question 18.
An aircraft executes a horizontal loop of radius km with a steady speed of 900 km h^_1 Compare its centripetal acceleration with the acceleration due to gravity
Answer:



Question 19.
Read each statement below carefully and state, with reasons, if it is true or false:
(a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the center.
(b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point.
(c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector.
Answer:
(a) The statement is false since the centripetal acceleration is towards the center only in the case of uniform circular motion  (constant speed) for which it is true.

(b) True, the velocity of a particle is always the tangent to the path of the particle at the point either in rectilinear or circular or curvilinear motion.

(c) True, because the direction of acceleration vector is always changing with time, always being directed towards the center  of the path followed in the uniform circular motion, so the resultant of all these vectors will be a null vector.

Question 20.
The position of a particle is given by r = 3.0 t-2.0t² +4.0 m
where t is in seconds and the coefficients have the proper units for r to be in meters.
(a) Find the  and  of the particle?
(b) What is the magnitude and direction of velocity of the particle at t = 2.0 s?=
Answer:
(a) Velocity



Question 21.
A particle starts from the origin at t = 0 s with a velocity of 10.0 m/s and moves in the x-y plane with a constant acceleration of (8.0  + 2.0 )
ms^-2.
(a) At what time is the x-coordinate of the particle 16 m? What is the y-coordinate of the particle at that time ?
(b) What is the speed of the particle at the time?
Answer:

Question 22.
 and  are unit vectors along x and y-axis respectively. What is the magnitude and direction of the vectors  +  and  –  ? What are the components of a vector A = 2+ 3 along the directions of +hat {j} [/latex] and  – 
Answer:



Question 23.
For any arbitrary motion in space, which of the following relations are true:

(The’average’stands for average of the quantity over the time interval t₁ to t₂)
Answer:
Relations (b) and (e) are true for any arbitrary motion in space. Relations (a), (c) and (d) are false as they hold for uniformly accelerated motion. For arbitrary motion, acceleration is not uniform.

Question 24.
Read each statement below carefully and state, with reasons and examples, if it is true or false:
A scalar quantity is one that
(a) is conserved in a process
(b) can never take negative values
(c) must be dimensionless
(d) does not vary from one point to another in space
(e) has the same value for observers with different orientations of axes.
Answer:
(a) The statement is false, as several scalar quantities are not conserved in a process.
For example energy being a scalar quantity is not conserved during inelastic collisions.

(b) The statement is false, because there are some scalar quantities which can be negative in a process.
For example, temperature being scalar quantity can be negative (-30°C, -4°C), charge being scalar can also be negative.

(c) The statement is false, there are large number of scalar quantities which may not be dimensionless.
For example, mass, density, charge etc. being scalar quantities have dimensions.

(d) The statement is false as there are some scalar quantities which vary from one point to another in space.
For example, temperature, gravitational potential, density of a fluid or anisotropic medium, charge density vary from point to point.

(e) The statement is true, orientation of axes does not change the value of a scalar quantity.
For example, mass is independent of the coordinate axes.

Question 25.
An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at a ground observation point by the aircraft positions 10.0 s apart is 30°, what is the speed of the aircraft?
Answer:
Suppose that O is the observation point on the ground. The aircraft is flying along XY at a height OC = 3,400 m from the ground. Let A and B be two positions of the aircraft 10 s apart. Thus, the aircraft goes from the point A to C (or from the point C to B) in 5 s. If the angle subtended by AB is 30° at the point O, then the angle subtended by AC (distance covered in 5 s) at O is 15°
From the right angled Δ OAC.

Question 26.
A vector has magnitude and direction. Does it have a location in space? Can it vary with time? Will two equal vectors a and b at different locations in space necessarily have identical physical effects? Give examples in support of your answer.
Answer:

1. A vector in general has no definite location in space because a vector remains unaffected whenever it is displaced anywhere in space provided its magnitude and direction do not change. However a position vector has a definite location in space.
2. A vector can vary with time e.g. the velocity vector of an accelerated particle varies with time
3. Two equal vectors at different locations in space do not necessarily have same physical effects. For example, two equal forces acting at two different points on a body which can cause the rotation of a body about an axis will not produce equal turning effect.

Question 27.
A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector?
Answer:
Generally, rotation is not considered a vector, despite the fact that it has the magnitude and direction. The reason is that the addition of two finite rotations does not obey the commutative law. Since addition of vectors should obey the commutative law, a finite rotation cannot be regarded as a vector, However, infinitesimally small rotations obey the commutative law for addition and hence an infinitesimally small rotation is a vector.

Question 28.
Can you associate vectors with
(a) the length of a wire bent into a loop,
(b) a plane area,
(c) a sphere? Explain.
Answer:
(a) No, we cannot associate a vector with the length of the wire bent into a loop.
(b) Yes, we can associate a vector with a plane area. The area vector is directed along normal to the plane area.
(c) No, we cannot associate a vector with a sphere.

Question 29.
A bullet fired at an angle of 30° with the horizontal hits the ground 3.0 km away. By adjusting its angle of projection, can one hope to hit a target 5.0 km away? Assume the muzzle speed to be fixed, and neglect air resistance.
Answer:

Question 30.
A fighter plane flying horizontally at an altitude of 1.5 km with speed 720 km h^-1 passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed 600 m s^-1 to hit the plane? At what minimum altitude should the pilot fly the plane to avoid being hit? (Take g=10m s^-2)
Answer:
Suppose that the fighter plane is flying horizontally with a speed υ at the height OA = 1.5 km. The point O represents the position of the anti-aircraft gun.

Let u be the velocity of the shell and 0, its inclination with the vertical. The shell hits the fighter plane at the point B as shown in Fig. Suppose that the shell hits the plane after a time f. Then, the horizontal distance travelled by the fighter plane in time t with velocity v is equal to the horizontal distance covered by the shell in time t with u_x, the r-component of its velocity i.e.

Question 31.
A cyclist is riding with a speed of 27 km h^-1. As he approaches a circular turn on the road of radius 80 m, he applies brakes and reduces his speed at the constant rate of 0.5 m s^-1 every second. What is the magnitude and direction of the net acceleration of the cyclist on the circular turn?
Answer:
Here, υ=27 km h^-1 = 7.5 m s^-1; r = 80 m Centripetal acceleration,

Suppose that the cyclist applies brakes at the point A of the circular turn. Then, retardation produced due to the brakes, say a_T will act opposite to the velocity, υ figure.

Free Projectile Motion Calculator – calculate projectile motion step by step.

Question 32.
(a) Show that for a projectile the angle between the velocity and the x-axis as a function of time is given by

(b) Shows that the projection angle θ_O for a projectile launched from the origin is given by

where the symbols have their usual meaning.
Answer:
(a) Let υ_ox and υ_oy be the initial component velocity of the projectile at O along OX direction and OY direction respectively, where OX is horizontal and the OY is vertical. Let the projectile go from O to P in time t and υ_x υ_y be the component velocity of projectile at P along horizontal and vertical directions respectively. Then, υ_y = υ_oy– gt and υ_x = υ_ox If 0 is the angle which the resultant velocity  makes with horizontal direction, then

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