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## Homework Statement

Hey guys, I'm doing a problem on quadratic drag and energy dissipation. Basically, the question asks me to find the energy dissipated when a ball of mass 'm' is thrown directly upwards with a velocity 'v

_{0}', during the upwards journey to its maximum height, 'h'. The resistive force is 'av

^{2}'. Now, whether or not it is relevant to answering the question (I think it is), I believe I need to find the integral of

'av

^{2}' with respect to the displacement upwards, 'x'.

## Homework Equations

I've derived the equation taking upwards as x-positive:

F = mvdv/dx = -mg -av

^{2}

ignoring any '-' signs,

E=∫ F dx = ∫( -mg -av

^{2})dx between x=0 and x=h

## The Attempt at a Solution

Now obviously the energy has to equal 1/2*m*v

_{0}

^{2}due to the conservation of energy.

And this can be easily shown by integrating the RHS - mvdv/dx.

But when I get to the av

^{2}, I tried substituting v=dx/dt, which doesn't make it any clearer.

I tried to do integration by parts, getting the av

^{2}part to be axv

^{2}- ∫2axvdv/dx dx.

Any ideas? Perhaps a knowledge of multi-variable calculus is needed?

**EDIT: Thanks for your ideas guys, but I *DON'T* need to find the height 'h' (which is what people are offering suggestions about). Essentially, I'd like to know how to do ∫av**

^{2}dx, with v being dx/dt:i.e. how do you find ∫ (dx/dt)

^{2}dx ??????

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