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jesuslovesu
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[SOLVED] Projectile Motion
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.
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.
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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