If, for a given velocity, the maximum range is at a projection angle of 45, then there must be equal ranges for angles above and below this. Show this explicitly.

Explanation:

The range R of a projectile is given by

The maximum range occurs when . Let be the angle above or below 45°. Now let's look at the ranges brought about by these angle differences.

Case 1: Angle above 45°

We can write the range as

Case 2: Angle below 45°

We can write the range as

Note that the equations (1) and (2) are identical. Therefore, the ranges are equal if they differ from 45° by the same amount.

as we know our atmosphere or air have different levels. every level has other gasses, and every gas has another permeability. when the light comes from lower permeability to higher permeability, it changes its direction. so that means that when the light comes from the moon to the earth, it changes it's direction many more times. it basically means that the shape of the light part of the moon is changed on the earth.

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