B Relationship between launch height and range

In a projectile motion practical, we had to investigate the relationship between launch of projectile and the range it travels. Increasing the height will obviously increase the range. However, we are trying to understand why it does in a more scientific way. We already know that increasing height means the projectile will have greater distance to cover vertically, and therefore the horizontal velocity will have more time to act upon the projectile, hence increasing the range. However, this is too 'non-scientific' for our purposes. We need a much more in-depth explanation as to why this occurs.

Thanks in advance,,
Stan
 

A.T.

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We already know that increasing height means the projectile will have greater distance to cover vertically, and therefore the horizontal velocity will have more time to act upon the projectile, hence increasing the range.
That assumes the same launch angle and thus same horizontal velocity. But for greater launch height the optimal angle is smaller, because you get more range by trading some vertical velocity (flight time) for horizontal velocity.

However, we are trying to understand why it does in a more scientific way.
Then you have to do the math.
 

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