The range of a projectile

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Question from Claudette, a parent:

This is a maximum minimum problem that my textbook didn't even try to give an example of how to do it in the text itself. It just suddenly appears in the exercises.

Problem: The range of a projectile is R = v^2 Sin 2x/g, where v is its initial velocity, g is the acceleration due to gravity and is a constant, and x is the firing angle. Find the angle that maximizes the projectile's range.

The author gives no information other than the formula.

I thought to find the derivative of the formula setting that to zero, but once I had done that, I still had nothing that addressed the author's question.

Any help would be sincerely appreciated.

Claudette

Claudette,

A unit analysis tells me that the formula should be

R = (v² Sin (2x) ) / g

otherwise you will not end up with a distance, which is what range represents. Perhaps that is a typo in the text, or that is what you meant to put in.

You should be able to see that the value of v does not affect the angle at which R is maximized. Rather, R is maximized when Sin(2x) is maximized. We know that the maximum value of sine is +1.

For what value x is sin(2x) = 1?

Cheers,
Stephen La Rocque.

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