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 'v0', during the upwards journey to its maximum height, 'h'. The resistive force is 'av2'. Now, whether or not it is relevant to answering the question (I think it is), I believe I need to find the integral of
'av2' with respect to the displacement upwards, 'x'.
I've derived the equation taking upwards as x-positive:
F = mvdv/dx = -mg -av2
ignoring any '-' signs,
E=∫ F dx = ∫( -mg -av2 )dx between x=0 and x=h
The Attempt at a Solution
Now obviously the energy has to equal 1/2*m*v02 due to the conservation of energy.
And this can be easily shown by integrating the RHS - mvdv/dx.
But when I get to the av2, I tried substituting v=dx/dt, which doesn't make it any clearer.
I tried to do integration by parts, getting the av2 part to be axv2 - ∫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 ∫av2 dx, with v being dx/dt:
i.e. how do you find ∫ (dx/dt)2 dx ??????