The maximum range of a projectile is the farthest distance that it can travel before returning to the ground, and it is determined by a variety of factors, including the initial velocity of the projectile, the angle at which it is launched, and the strength and direction of the gravitational force acting upon it.

One of the key factors that determines the maximum range of a projectile is the angle at which it is launched. When a projectile is launched at an angle, it is subject to both the gravitational force pulling it towards the ground and the horizontal velocity that it was given at launch. These two forces act on the projectile simultaneously, causing it to follow a parabolic path through the air.

The angle at which a projectile is launched plays a crucial role in determining its maximum range because it determines the balance between the horizontal and vertical components of its motion. If a projectile is launched at a low angle, it will have a large horizontal velocity but a small vertical velocity, and will therefore travel a long distance horizontally before hitting the ground. On the other hand, if a projectile is launched at a high angle, it will have a large vertical velocity but a small horizontal velocity, and will therefore travel a shorter distance horizontally before hitting the ground.

It turns out that the angle that maximizes the range of a projectile is 45 degrees. This is because at this angle, the horizontal and vertical components of the projectile’s motion are equal, allowing it to travel the farthest distance before returning to the ground.

To understand why this is the case, consider the motion of the projectile at the highest point of its trajectory, when it is momentarily at rest. At this point, the projectile’s vertical velocity is zero, and all of its energy is in the form of kinetic energy. The kinetic energy of a projectile can be expressed as:

KE = (1/2) * m * v^2

where m is the mass of the projectile and v is its velocity.

Since the projectile is at the highest point of its trajectory when its vertical velocity is zero, the kinetic energy at this point is entirely due to its horizontal velocity. Therefore, the horizontal velocity of the projectile at the highest point of its trajectory is a measure of its maximum range.

If the projectile is launched at an angle other than 45 degrees, the horizontal velocity at the highest point of its trajectory will not be maximal, resulting in a shorter range. On the other hand, if the projectile is launched at an angle of 45 degrees, the horizontal velocity at the highest point of its trajectory will be at its maximum, resulting in the maximum range.

In summary, the maximum range of a projectile is achieved when it is launched at an angle of 45 degrees because at this angle, the horizontal and vertical components of its motion are equal, allowing it to travel the farthest distance before returning to the ground.