Question

A ball is thrown from the rear end of the compartment of the train to the front which is moving at a constant horizontal velocity. An observer A sitting in the compartment and another observer B standing on the ground drew the trajectory. They will have
A. equal horizontal and equal vertical ranges.
B. equal vertical ranges but different horizontal ranges.
C. different vertical ranges but equal horizontal ranges.
D. different vertical ranges and different horizontal ranges.

Answer 1

Use the formula for the range of the projectile. This formula gives the relation between the range, initial velocity, angle of projection and acceleration due to gravity.

Formula used:
The horizontal range $R$ of the projectile is given by
$R = \dfrac{{{u^2}\sin 2\theta }}{g}$ …… (1)
Here, $u$ is the initial velocity of the projectile, $\theta$ is the angle of projection and $g$ is the acceleration due to gravity.

Complete step by step answer:
The horizontal component of velocity of projection of ball for the observer in the compartment and the observer standing on the ground is different.Hence, the observers will not have equal horizontal ranges.Hence, the option A is incorrect.

The vertical component of velocity of the projection of the ball is zero for both observers. Hence, they will have the same vertical ranges.Hence, the option C is incorrect.

From the above explanations, both the observers will have the same vertical range and different horizontal range.Hence, the option D is incorrect.

From equation (1), it can be concluded that the range of the projectile depends on initial velocity of projection, angle of projection and acceleration due to gravity.Since angle of projection and acceleration due to gravity remains the same for both A and B, range will depend only on the initial velocity.The vertical component of the initial velocity of projection is zero as the ball is moving with horizontal velocity. Hence, both the observers A and B will draw trajectories with equal vertical range.

But the horizontal component of velocity of projection is not the same for observers A and B. For observer A, the initial horizontal velocity component will be equal to the speed at which the ball is thrown with respect to the train. For observer B, the velocity of the ball will be the resultant of the velocity of the ball which is equal to velocity of train and velocity of the ball with respect to ground. Hence, the observers A and B will draw trajectories with different horizontal ranges.Therefore, the vertical ranges for both the observers are the same but horizontal ranges are different.

Hence, the option B is correct.

Note: The formula for the ranges includes only initial velocity. Hence, we have considered only vertical and horizontal components of initial velocity. Also the vertical and horizontal and vertical components of velocity of ball at any time cannot be determined only from the given information, hence not considered in solution.