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**[SOLVED] Projectile Motion**

## Homework Statement

A cannon shoots a ball at an angle theta above the ground. Use Newton's second law to find the ball's position as a function of time g(t). What is the largest value of theta if g(t) is to increase throughout the ball's flight.

## Homework Equations

## The Attempt at a Solution

Alright so I've found the distance squared (as advised in the original problem).

[tex]g(t) = r^2 = 1/4g^2 t^4 - (v_0 g sin \theta ) t^3 + {v_0}^2 t^2[/tex]

Which I have verified is correct.

So then like usual, I differentiate g(t) with respect to time to find the critical points.

[tex]g' = g^2 t^3 - 3(v_0 g sin \theta t^2 + 2 {v_0}^2 t[/tex]

Then I use the quadratic formula to find t and set that equal to 0

[tex]\frac{3v_0 g sin \theta +- \sqrt{9 {v_0}^2 g^2 sin^2 \theta - 8g^2 {v_0}^2} }{2g^2}[/tex]

This is where the problem comes in...

If I were to set g' equal to 0 and solve

[tex]0 = 3v_0 g sin \theta +- \sqrt{9 {v_0}^2 g^2 sin^2 \theta - 8g^2 {v_0}^2} [/tex]

But then if I solve to 0 I get [tex]8g^2 {v_0}^2 = 0[/tex]

I know the answer involves setting the discriminant equal to 0, but I don't understand why I can't find the critical points.

So basically my question is how to find theta max? Generally I would find the critical points and then determine if it's a max or a min. In this case, it doesn't seem to work.

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