Physics Friday 114

Consider a semicircle formed from n distinct, identical, equally spaced masses, with total mass M. Now, let us send in a mass m with initial speed v toward one end of the semicircle, so that it undergoes successive perfectly elastic collisions with each and every one of the n masses. What restrictions are there on m that this is possible, and what is the total momentum of the n masses after the collisions in the limiting case on m? In particular, what are both of these in the limit of large n?

Note that after all n collisions, the mass m will have been deflected by a total angle π, by symmetry (or via geometry), we see then that each collision must deflect this mass by an angle of . Note that each of the masses of our semicircle, the deflecting masses, must have mass . We found here the maximum angle by which a larger mass can be deflected by a smaller one in a perfectly elastic collision, giving us , and thus, we see
and thus
with equality being our limiting case. When n is large, θ is small, so , and with this approximation,

Now, we found here that in the maximum angle deflection case, the velocity post-collision has magnitude ,
and recall that in the mass limit, we have , so after our first collision, mass m has speed

each collision is similar, so each reduces the velocity by the factor , and so the final speed of the mass m after all the collisions is
in a direction opposite that of the original velocity; thus, the momentum has undergone a change of , which is equal to the total combined momentum imparted to the n masses.
For small θ, we see that to first order,
, giving us a large n approximation of
and recalling that , we see that in the large n limit,
and so the net momentum transfer to the masses is


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