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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