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succession of elastic collisions?
a ball of mass m ( initially at rest on a frictionless surface) is hit by a succession of small particles of mass δm and velocity v0 in the positive x direction. Note that δm<<m and each particle δm rebounds in the negative x-direction. We're asked to show that the speed,v, of the ball m after the nth collision is approximately given by
v=v0[1- exp(-an)] where a=2δm/m.
I tried this for the first collision with δm and found that the speed of m is given by
2δmv0/(m+δm)
but now I'm not sure what to do next (to get n in the equation). I have a feeling I have to integrate over something but I'm really not sure.
Any help would be much appreciated!
Thanks.
I'm trying to find the speed so integrating wrt time wouldn't help. Also, the expression I found for speed is presumably valid for the speed after the first collision only because the mass m will not be stationary for the second collision (hence the equations for the conservation of momentum and kinetic energy will be slightly different.......... unless we neglect this increase in energy?)
