- #1

PhizKid

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## Homework Statement

I'm only interested in the second part where V << v

## Homework Equations

Energy

## The Attempt at a Solution

First I boosted to the frame of the moving wall to find the change in velocity of the ball after a collision with it. In the frame of the moving wall, the ball first comes towards it with some velocity [itex]v + V[/itex] and leaves with some velocity [itex]v' - V[/itex] so by conservation of momentum, [itex]v + V = v' - V[/itex] (I don't need to insert a negative sign on the left hand side expression right since I already took into account the direction by indicating its velocity would increase by V in the frame of the wall?) therefore [itex]\Delta v = 2V[/itex]. Now focusing on the stationary wall, let's say the ball bounces off the wall and travels a distance [itex]x[/itex] to the moving wall and then of course travels the same distance x back to the stationary wall. The time for this round trip is [itex]\Delta t = \frac{x}{v} + \frac{x}{v + 2V} = \frac{2(vx + Vx)}{v^{2} + 2vV} = \frac{2vx(1 + \frac{V}{v})}{v^{2}(1 + 2\frac{V}{v})} = \frac{2x(1 + \frac{V}{v})(1 + 2\frac{V}{v})^{-1}}{v} \approx \frac{2x}{v}(1 + \frac{V}{v})(1 - 2\frac{V}{v}) = \frac{2x}{v}(1 - \frac{V}{v} - 2(\frac{V}{v})^{2}) \approx \frac{2x}{v} + O(\frac{V}{v})[/itex] where I have used a binomial expansion. Then we have that [itex]\lim_{\Delta t\rightarrow 0}\frac{\Delta v}{\Delta t} = \frac{\mathrm{d} v}{\mathrm{d} t} = \frac{vV}{x} = \frac{vV}{l - Vt}[/itex] where the last part comes from the diagram. Integrating this and applying initial conditions gives [itex]v = \frac{v_{0}l}{l - Vt} = \frac{v_{0}l}{x}[/itex]. The average force on the wall is [itex]-\frac{\Delta p}{\Delta t} = \frac{2mv}{\Delta t} = \frac{mv^{2}}{x} = \frac{m(\frac{v_{0^{2}l^{2}}}{x^{2}})}{x} = \frac{mv_{0}^{2}l^{2}}{x^{3}} = \frac{mv_{0}^{2}}{l}(\frac{l}{x})^{3}[/itex]. This agrees with the book but I have three questions: was the way I found the change in velocity correct (like I said in the parenthesis at the start of the post I'm unsure about the signs), I made many approximations is that ok, and finally this is the average force on the stationary wall but the problem doesn't specify which one, will the one on the moving wall be the same? Thanks!