# Thread: multi-variable ballistic optimization

1. ## multi-variable ballistic optimization

A material point slides up a ramp of length l making an angle α = 30° with the horizontal. Its initial speed (at the bottom of the ramp) is 10m/s. This point is subject to gravity with g = 9.81m/s². At the top of the ramp the material point starts a free fall and falls on the ground with a reach The goal is to get the ramp length that maximizes that reach. What is this length?

I don't know how to begin
Any suggestions ?

2. ## Re: multi-variable ballistic optimization

You posted this under "Advanced Algebra" but there is no "Advanced Algebra" here. (Nor is it "mult-variable. The only variable is l. The rest are constants.) I would treat it as a Calculus problem- take the derivative of the function with respect to l, set that equal to 0, and solve for l.

3. ## Re: multi-variable ballistic optimization Originally Posted by HallsofIvy Then do what I presume you learned in Calculus- take the derivative with respect to l, set it equal to 0, and solve for l.
only problem with that is that this is the advanced algebra forum....

4. ## Re: multi-variable ballistic optimization

Yes, but there was no "Advanced Algebra" in the problem! I have edited my response.

5. ## Re: multi-variable ballistic optimization Originally Posted by HallsofIvy Yes, but there was no "Advanced Algebra" in the problem! I have edited my response.
Ha! That is one way to handle it.

Using calculus it can be shown that the optimal length of the ramp is twice the x coordinate of the vertex of the quadratic in the radical.

That is $\ell = \dfrac{2v_0^2}{3g}$