Projectile Motion

The ball is projected horizontally, so the horizontal velocity does not affect the time taken to hit the ground.
The time to fall depends only on the vertical motion, which is influenced by gravity.
The equation for the time taken for an object to fall freely from a height is: where:
- is the height of the building,
- is the acceleration due to gravity. Substituting the values into the equation:
Thus, the time taken for the ball to hit the ground is . Therefore, the correct answer is:

To determine the kinetic energy at the highest point of the projectile's motion, we need to consider both the initial kinetic energy and the nature of projectile motion. When a particle is projected at an angle with initial kinetic energy , the initial velocity components are:

• Horizontal component:
• Vertical component:

At the highest point of its trajectory, the vertical component of velocity becomes zero, while the horizontal component remains unchanged due to the absence of horizontal forces (assuming air resistance is negligible).

Therefore, the kinetic energy at the highest point depends only on this horizontal component. Initial kinetic energy, , is given by:

The horizontal kinetic energy at the highest point is:

Substituting and , we have:

Substitute :

This shows that the kinetic energy at the highest point is .

The maximum height attained by a projectile is given by the formula: where: - is the initial velocity, - is the angle of projection, - is the acceleration due to gravity.
Case 1: Original values In the first case, let the initial velocity be and the angle of projection be . The maximum height attained in the first case is:
Case 2: Doubled mass and angle In the second case, the mass is doubled, but the initial velocity remains the same. The angle of projection is also doubled, so the angle becomes . The maximum height in the second case is: Using the double angle identity for sine, we know that:
Thus, the maximum height in the second case becomes:
Now, the ratio of the maximum heights in the first case to the second case is: For , , so:
Thus, the ratio of the maximum heights attained is , and the correct answer is option (C).