# A and B are two trains moving parallel to each other. If a balls is thrown vertically up from the...

## Question:

{eq}A {/eq} and {eq}B {/eq} are two trains moving parallel to each other. If a balls is thrown vertically up from the train {eq}A {/eq}, the path of the ball is:

a. Parabola for an observer standing on the ground,

b. Vertical straight line for an observer in {eq}B {/eq} when {eq}B {/eq} is moving with same speed but in same direction,

c. A parabola for an observer in {eq}B {/eq} when {eq}B {/eq} is moving with same speed but in opposite direction,

d. All the above are true.

## Inertial Frame of Reference

A frame of reference associated with an object moving with constant velocity is called an inertial frame of reference. Newton's laws will give the correct dynamics in an inertial frame. Ideally, a frame of reference attached to a distant fixed star is taken as the inertial frame. If the frame is accelerated, then immediately pseudo forces come into play. If these are incorporated then Newton's laws can again give the correct dynamics.

## Answer and Explanation:

If a particle is projected near the surface of the earth at an angle to the horizontal then it will trace out a parabola. If projected vertically then the path is a straight line. Thus a horizontal velocity component is necessary for generating the parabolic trajectory.

Here it is given that there are two trains A and B moving along parallel tracks with constant velocity. So the frames of reference are inertial. A ball is thrown vertically up in train A. Clearly in the train frame of reference the ball has no horizontal velocity component. Therefore it will go straight up and then come back straight down.

For a ground-based observer, the ball has an initial horizontal velocity component the same as that of the train. So in the ground frame, the trajectory is a parabola.

If train B is moving parallel to A with the same speed then the ball has no horizontal velocity component in the B frame. Therefore the path is a vertical straight line.

If B is moving in the opposite direction then in the B frame the ball does have a horizontal velocity. Therefore the trajectory is a parabola.

Thus all the given statements are true.

The correct answer is **Option d)**.

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Chapter 4 / Lesson 12