Answer:

To solve this question, we need to understand the relationship between the initial speed and the speed at the maximum height of a projectile. Let's break down the problem step by step.

Step 1: Understand the motion of a projectile
A projectile is an object that is launched into the air and moves along a curved path under the influence of gravity. The motion of a projectile can be divided into two components: horizontal and vertical.

Step 2: Analyze the vertical motion
In the vertical direction, the projectile follows a parabolic path. At the highest point of its trajectory, the vertical velocity becomes zero. This means that the speed of the projectile at its maximum height is zero.

Step 3: Analyze the horizontal motion
In the horizontal direction, the projectile moves with a constant velocity. The horizontal velocity remains unchanged throughout the motion. Therefore, the speed of the projectile at its maximum height is equal to its horizontal velocity.

Step 4: Relate the initial speed and the speed at the maximum height
Since the speed of the projectile at its maximum height is equal to its horizontal velocity, we can conclude that the speed at the maximum height is half of the initial speed when the projectile is launched at an angle of 45 degrees.

Step 5: Determine the angle of projection
Given that the speed at the maximum height is half of the initial speed, we can conclude that the angle of projection is 45 degrees.

Final Answer: The angle of projection is 45 degrees (option 4).