## Ball bouncing from massive wall.Most physics textbooks consider the case of a ball bouncing from a massive object, say the floor, or a wall. They consider the case where the collision is nearly or totally elastic. In the totally elastic collision, the ball loses no kinetic energy in the collision, so its speed after collision is the same as before the collision. The student thinks, "Of course that must be the case, because of conservation of energy." Seldom does the textbook, or the student, consider how conservation of momentum is satisfied in this problem. They should, for the analysis is instructive. Analyzing this may also give some insight into why energy, momentum and the conservation laws took so long to be formulated, since the concepts are subtle. Consider a ball of mass m colliding elastically with a stationary object of larger mass M. Draw the picture before and after the collision. The conservation equations are:
m v + M_{2}V _{2}
(1/2)mv where v Multiply the energy equation by 2 to eliminate the (1/2) factors.
mv Divide this by m on both sides.
v Rearrange.
v Divide the momentum equation by m on both sides.
v Rearrange and square both sides.
(v Multiply Eq. [5] by (M/m) and combine with [7] to eliminate V
(M/m)(v Multiply both sides by (m/M)
(v Take the limit as (m/M) goes to zero.
(v So one solution of this is v One can graph the values of v Those who have had calculus will recognize that this is a case where an indeterminate form arises when you take the limit of the value of momentum of a body whose mass increases to infinity. ## Bouncing at an angle.A related problem is that of a ball bouncing elastically from a massive wall incident at an angle. If the ball is incident to the wall at some angle, the rebound is at the same angle, provided the ball had no spin before or after the collision. That's actually a rather unlikely situation, for even if the ball had no spin initially, it usually does after the collision and the incident and rebound angles will not be equal. The problem can therefore get quite messy. But in the textbook discussions of classical kinetic theory one often begins with an idealized situation of - —Donald E. Simanek, March 5, 2005
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