Question

Two projectiles are projected simultaneously from the top of a tower. $A$ is dropped and $B$ is thrown horizontally. For this situation mark out the correct statement(s).

• A. Both reach the ground simultaneously
• B. Both reach the ground with same speed
• C. Both reach the ground with different speed
• D. Both have the same vertical displacement when they reach the ground.

Answer 1

Answer: (A), (C), (D)

Both have the same vertical displacement when they reach the ground.

Diagrammatic representation of A's motion:

Here, $u$ is the initial velocity of the ball, $a$ is the acceleration acting on the ball and $g$ is the acceleration due to gravity.

Diagrammatic representation of B's motion:

Here, $u_x$ and $u_y$ are the horizontal and vertical components of the initial velocity of the ball, $a$ is the acceleration acting on the ball and $g$ is the acceleration due to gravity.

Explanation for correct option:

Option A:

1. Ball A is falling freely from the tower with an initial velocity of zero and its direction is downwards.
2. Ball B is moving in a projectile motion with the initial velocity equal to the horizontal component of the velocity.
3. The vertical component of the velocity of ball B is zero.
4. Both the balls will reach the ground simultaneously because the vertical velocities of both the balls are zero and their directions are also the same.

Option C:

1. The formula to calculate the velocity vector at any instant is $\mathit{v}\mathbf{=}\sqrt{{\mathbf{\left(}{\mathbf{v}}_{\mathbf{x}}\mathbf{\right)}}^{\mathbf{2}}\mathbf{+}{\mathbf{\left(}{\mathbf{v}}_{\mathbf{y}}\mathbf{\right)}}^{\mathbf{2}}}$
2. Both reach the ground at different speeds because $B$ has both horizontal and vertical velocity while $A$ only has vertical velocity when it reaches the ground.

Option D:

1. The formula to calculate the vertical displacement is ${\mathit{s}}_{\mathbf{y}}\mathbf{=}{\mathit{u}}_{\mathbf{y}}\mathit{t}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathit{a}{\mathit{t}}^{\mathbf{2}}$
2. The vertical components of their initial velocities are the same.
3. The acceleration acting on both the balls is equal to the acceleration of the gravity.
4. Both the balls reach the ground at same time.
5. Hence, both will have the same vertical displacement when they reach the ground.

Explanation for incorrect options:

Option B:

Both did not reach the ground with the same speed as $B$ has both horizontal and vertical velocity while $A$ has only vertical velocity.

Hence, options A, C, and D are the correct options.