Question

Assertion: The two projectiles having same time of flight would have same range.

Reason: The two projectiles having same time of flight would have same maximum height.

• A. both assertion and reason are true and reason is the correct explanation of assertion.
• B. both assertion and reason are true but the reason is not the correct explanation of assertion.
• C. assertion is true but reason is false.
• D. assertion is false but the reason is true.

Answer 1

The correct option is D

assertion is false but the reason is true.

Projectile motion:

The diagrammatic representation of the projectile motion of a projectile is given below:

1. Here, $u$ is the velocity of the projectile, $u_x$ is the horizontal component and $u_y$ is the vertical component of the velocity of the projectile at an instant.
2. Here, $\theta$ is the angle of projection and $A$ is the point where the projectile has the maximum height.
3. The only acceleration acting on the projectile is gravity. Hence, the direction of acceleration will be downwards. The downward direction is considered a negative direction here.

Explanation for assertion statement:

1. The formula of the time of flight is $\mathit{T}\mathbf{=}\frac{\mathbf{2}{\mathbf{u}}_{\mathbf{y}}}{\mathbf{g}}$.
2. If the time of flight of two projectiles are the same it will mean that both projectiles are having same vertical velocities at that instant.
3. This does not mean that their initial velocities are the same.
4. The formula of the range of the projectile is $\mathit{R}\mathbf{=}\frac{{\mathbf{u}}^{\mathbf{2}}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{2}\mathbf{\theta }}{\mathbf{g}}$.
5. In order to have the same range, the two projectiles need to have the same initial velocities as well as angles of projection with the horizontal.
6. Hence, the assertion is not true.

Explanation for reason statement:

1. The formula of the maximum height is $\mathit{H}\mathbf{=}\frac{{{\mathbf{u}}_{\mathbf{y}}}^{\mathbf{2}}\mathbf{}}{\mathbf{2}\mathbf{g}}$.
2. From the above formula, it is clear that the maximum height is only dependent upon the vertical component of the velocity of the projectiles at an instant.
3. It is given that the time of flight of the two projectiles are the same. Hence, both the projectiles are having same vertical velocities at that instant.
4. Therefore, the maximum heights of both are the same
5. Hence, the reason is true.

Hence, option D is the correct answer.