Question

The two projectiles would have same maximum height if their

• A. time periods of flight are the same
• B. ranges are the same
• C. vertical components of initial velocity are the same
• D. Horizontal components of initial velocity are the same

Answer 1

Answer: (A), (C)

vertical components of initial velocity are the same

Maximum Height::

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.
4. In projectile motion, the value of the vertical component of the velocity becomes zero at the highest point because acceleration was acting on it in the opposite direction till it reaches the maximum height.
5. The value of the horizontal component of the velocity remains constant throughout the projectile motion.

Explanation for correct options:

Option A:

1. The formula to calculate maximum height is $\frac{{\left(u_y\right)}^{\mathbf{2}}}{\mathbf{g}}$
2. The value of gravity remains the same for all the projectile motions. Thus, for the same maximum height same vertical component of the velocity is required.
3. The formula to calculate the time of flight is $\frac{{\mathbf{u}}_{\mathbf{y}}}{\mathbf{2}\mathbf{g}}$. From the formula, it is clear that it is also dependent on the value of $u_y$ only.
4. $u_y$ is same only if the sum of both the angles of projections of both the projectiles is $90°$
5. Thus, if the value of $u_y$ is same then the value of the time of flight will be the same for both the projectiles.
6. Hence, the two projectiles would have the same maximum height if their time of flight are the same.

Option C:

The two projectiles would have the same maximum height if their vertical components of initial velocity are the same because maximum height is only dependent on the $u_y$ of the projectile.

Explanation for incorrect options:

Option B:

1. The formula to calculate range is $\frac{{\mathbf{u}}^{\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathbf{2}\mathit{\theta }}{\mathbf{g}}$.
2. The maximum height can be the same if the vertical components of the initial velocity of the projectiles are the same because the formula to calculate maximum height is $\frac{{\left(u_y\right)}^{\mathbf{2}}}{\mathbf{g}}$.
3. As equal ranges do not that the vertical components of the initial velocity of the projectiles are the same.
4. Hence, option B is an incorrect option.

Option D:

1. The maximum height is only dependent on the $u_y$ of the projectile.
2. Hence, option D is an incorrect option.

Hence, options A and C are correct.