Answers to Problems on (Kinematics - Rest and Motion) HC Verma's Questions for Short Answer (8-14)

 Q#8

If a particle is accelerating, it is either speeding up or speeding down. Do you agree with this statement?

Answer:
This statement cannot be agreed upon. Consider the case of a uniform circular motion where acceleration is always perpendicular to the path of motion. Here the particle neither speeds up nor speeds down but moves with constant speed.

Q#9
A food packet is dropped from a plane going at an altitude of 100 m. What is the path of the packet as seen from the plane? What is the path as seen from the ground? If someone asks "What is the actual path", what will you answer? 

Answer:
The path of the packet as seen from the plane is a straight line. The path of the packet as seen from the ground is parabolic.

The actual path has no meaning, we can describe the path only with respect to some reference point.

Q#10
Give an example where (a) the velocity of a particle is zero but its acceleration is not zero, (b) the velocity is opposite in direction to the acceleration, (c) the velocity is perpendicular to the acceleration. 

Answer:
The followings are the examples:-

(a) A particle thrown vertically upwards has its velocity zero at the highest point but its acceleration is not zero. The acceleration is 'g' due to gravity vertically downwards.

(b) When a particle is thrown vertically upwards its velocity at any instant during upwards movement is opposite in direction to the acceleration. Velocity is vertically upwards while acceleration due to gravity is vertically downwards.

(c) In a uniform circular motion, the velocity is perpendicular to the acceleration.
  
Q#11
Figure (3-Q1) shows the x- coordinate of a particle as a function of time. Find the signs of v_xand a_xat t = t₁, t = t₂  and t = t₃.    

Answer:
If x' and x" be the x-coordinate of the particle at initial time t' and t" respectively then

v_x = (x"- x')/(t"-t') = tan θ.

For t"- t' infinitesimally small it is the vx at that instant.

So the slope of the tangent at any point in the above graph gives vx.

At t = t₁, tan θ is positive, so sign of v_x is positive. At t= t₂ the slope of the curve is horizontal, so

tan θ = 0 → v_x = 0. At t = t₃ the slope of the curve is negative, so sign of v_x is negative.

Sign of aₓ. Acceleration is the rate of change of velocity. For the sign of the acceleration at any instant, we see the slope of the curve (which is velocity) before and after the instant. If the slope after the instant is more than before the instant, it means the velocity is increasing i.e. the acceleration is positive, otherwise, it is just opposite.

At t = t₁, the slope is increasing. Hence the sign of aₓ is positive. At t = t₂, the slope is less after the instant than it was before, so the velocity is decreasing, hence the sign of aₓ is negative here. At t = t₃, the slope before the instant is less while after the instant it is more, so the velocity is increasing, hence the acceleration (aₓ) is positive.

Q#12
A player hits a baseball at some angle. The ball goes high up in the space. The player runs and catches the ball before it hits the ground. Which of the two (the player or the ball) has greater displacement?  

Answer:  
Both have equal displacements because both (the player and the ball)  start from the same point and stop to the same point. Even though their paths are different the displacement is the vector joining initial to the final position.

Q#13
The increase in the speed of a car is proportional to the additional petrol put into the engine. Is it possible to accelerate a car without putting more petrol or less petrol into the engine? 

Answer:
It is possible to accelerate a car without putting more petrol or less petrol into the engine if it is driven down a hilly road. The slope of the road should be more than enough to overcome the force of friction.

Q#14
Rain is falling vertically. A man running on the road keeps his umbrella tilted but a man standing on the street keeps his umbrella vertical to protect himself from the rain. But both of them keep their umbrella vertical to avoid the vertical sun-rays. Explain.

Answer:
The tilt of the umbrella will be in the direction of the relative velocity of the rain or sun-rays to the man. The relative velocity of the rain or sun-rays to the man will be the velocity of rain or sun-rays minus the velocity of man. For a standing man, this relative velocity will be vertical because his own velocity is zero so he keeps his umbrella vertical for both the rain or sun-rays. 

For the running man in the case of rain this relative velocity is at an angle to the vertical, so to protect himself he keeps his umbrella tilted in the direction of the relative velocity. In the case of sun-rays the velocity of light is so much greater that man's velocity is negligible. So the relative velocity of sun-rays is practically vertical so the running man keeps his umbrella vertical.

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