Introduction:
When a projectile is projected in the air at some angle with the horizontal, it experiences the force of gravity acting vertically downward. This force of gravity, also known as weight, can be used to determine the average power delivered by the weight of the projectile. Additionally, there are certain instances when the instantaneous power delivered by the weight of the projectile is zero. In this response, we will explore the average power delivered by weight of a projectile and analyze when the instantaneous power is zero.

Average Power Delivered by Weight of a Projectile:
The average power delivered by the weight of a projectile can be calculated using the formula:

Power = Work / Time

In the case of a projectile, the work done by the weight is equal to the change in gravitational potential energy. The gravitational potential energy can be calculated using the formula:

Gravitational Potential Energy = m * g * h

Where m is the mass of the projectile, g is the acceleration due to gravity, and h is the change in height.

To find the average power, we need to divide the change in gravitational potential energy by the time it takes for the projectile to reach that height. This can be expressed as:

Average Power = (m * g * h) / t

Where t is the time taken for the projectile to reach the height h.

Instantaneous Power:
The instantaneous power delivered by the weight of the projectile can be determined by analyzing the forces acting on the projectile at a particular instant in time. Since the weight of the projectile always acts vertically downward, the instantaneous power delivered by the weight is given by:

Instantaneous Power = Weight * Velocity * cosθ

Where Weight is the magnitude of the weight vector, Velocity is the magnitude of the velocity vector of the projectile, and θ is the angle between the weight vector and the velocity vector.

When is the Instantaneous Power Zero?
The instantaneous power delivered by the weight of the projectile is zero when the angle θ between the weight vector and the velocity vector is 90 degrees. In this case, the cosine of 90 degrees is zero, resulting in zero instantaneous power. This occurs when the projectile is moving horizontally, parallel to the ground.

When the projectile is at its highest point in the trajectory, the velocity vector is purely horizontal, and the weight vector is purely vertical. Therefore, the angle between them is 90 degrees, making the instantaneous power zero.

Summary:
The average power delivered by the weight of a projectile can be calculated using the change in gravitational potential energy over time. The instantaneous power delivered by the weight of the projectile is zero when the angle between the weight vector and the velocity vector is 90 degrees. This occurs when the projectile is at its highest point in the trajectory.