Chapter 4 Motion in a plane Solutions
Question - 21 : - 
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 
along the directions of 
and ? [You may use graphical method]
Answer - 21 : - 
Question - 22 : - For any arbitrary motion in space, which of the following relations are true:
Answer - 22 : -
Answer: (b) and (e)
(a)It is given that the motion of the particle is arbitrary. Therefore, the average velocity of the particle cannot be given by this equation.
(b)The arbitrary motion of the particle can be represented by this equation.
(c)The motion of the particle is arbitrary. The acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of the particle in space.
(d)The motion of the particle is arbitrary; acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of particle in space.
(e)The arbitrary motion of the particle can be represented by this equation.
Question - 23 : - 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 - 23 : -
(a) False, because kinetic energy is a scalar but does not remain conserved in an inelastic collision.
(b) False, because potential energy in a gravitational field may have negative values.
(c) False, because mass, length, time, speed, work etc., all have dimensions.
(d) False, because speed, energy etc., vary from point to point in space.
(e) True, because a scalar quantity will have the same value for observers with different orientations of axes since a scalar has no direction of its own.
Question - 24 : - 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 s apart is 30°, what is the speed of the aircraft? Time taken by aircraft from A to B is 10 s.
Answer - 24 : - The positions of the observer and the aircraft are shown in the given figure.
Height of the aircraft from ground, OR = 3400 m
Angle subtended between the positions, ∠POQ = 30°
Time = 10 s
In ΔPRO:
ΔPRO is similar to ΔRQO.
∴PR = RQ
PQ = PR + RQ
= 2PR = 2 × 3400 tan 15°
= 6800 × 0.268 = 1822.4 m
∴Speed of the aircraft 
A vector has magnitude and direction.
(i) Does it have a location in the space?
(ii) Can it vary with time?
(iii) 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 - 25 : -
(i) Besides having magnitude and direction, each vector has also a location in space.
(ii) A vector can vary with time. As an example, velocity and acceleration vectors may vary with time.
(iii) Two equal vectors a and b having different locations may not have same physical effect. As an example, two balls thrown with the same force, one from earth and the other from moon will attain different ‘maximum heights’.
Question - 26 : - A vector has both magnitude and direction. Does that mean 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 - 26 : - No. Finite rotation of a body about an axis is not a vector because finite rotations do not obey the laws of vector addition.
Question - 27 : - Can you associate vectors with (a) the length of a wire bent into a loop (b) a plane area (c) a sphere? Explain.
Answer - 27 : -
(a) We cannot associate a vector with the length of a wire bent into a loop. This is because the length of the loop does not have a definite direction.
(b) We can associate a vector with a plane area. Such a vector is called area vector and its direction is represented by a normal drawn outward to the area.
(c) The area of a sphere does not point in any difinite direction. However, we can associate a null vector with the area of the sphere. We cannot associate a vector with the volume of a sphere.
Question - 28 : - A bullet fired at an angle of 30° with the horizontal hits the ground 3 km away. By adjusting its angle of projection, can one hope to hit a target 5 km away? Assume the muzzle speed to the fixed, and neglect air resistance.
Answer - 28 : - 
Question - 29 : - A fighter plane flying horizontally at analtitude of 1.5 km with speed 720 km/h passes directly overhead ananti-aircraft gun. At what angle from the vertical should the gun be fired forthe shell with muzzle speed 600 m s–1 tohit the plane? At what minimum altitude should the pilot fly the plane to avoidbeing hit? (Take g =10 m s–2).
Answer - 29 : -
Height of the fighter plane = 1.5 km = 1500 m
Speed of the fighter plane, v = 720 km/h = 200 m/s
Let θ bethe angle with the vertical so that the shell hits the plane. The situation isshown in the given figure.
Muzzle velocity of the gun, u =600 m/s
Time taken by the shell to hit the plane = t
Horizontal distance travelled by the shell = uxt
Distance travelled by the plane = vt
The shellhits the plane. Hence, these two distances must be equal.
uxt = vt
In order to avoid being hit by the shell,the pilot must fly the plane at an altitude (H) higher than the maximum heightachieved by the shell.
Question - 30 : - A cyclist is riding with a speed of 27 km/h. 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 every second. What is the magnitude and direction of the net acceleration of the cyclist on the circular turn ?
Answer - 30 : - 