## 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**_{1} = m**v**_{2} + M**V**_{2}

(1/2)mv_{1}^{2} = (1/2)mv_{2}^{2} + (1/2)MV^{2}

where v_{1} is the initial velocity of the smaller ball, v_{2} is its final velocity after collision, and V_{2} is the velocity of the larger mass after the collision.

Multiply the energy equation by 2 to eliminate the (1/2) factors.

mv_{1}^{2} = mv_{2}^{2} + MV^{2}

Divide this by m on both sides.

v_{1}^{2} = v_{2}^{2} + (M/m)V_{2}^{2}

Rearrange.

v_{1}^{2} - v_{2}^{2} = (M/m)V_{2}^{2}

Divide the momentum equation by m on both sides.

v_{1} = v_{2} + (M/m)V_{2}

Rearrange and square both sides.

(v_{1} - v_{2})^{2} = (M/m)^{2}V_{2}^{2}

Multiply Eq. [5] by (M/m) and combine with [7] to eliminate V_{2}^{2}.

(M/m)(v_{1}^{2} - v_{2}^{2}) = (v_{1} - v_{2})^{2}

Multiply both sides by (m/M)

(v_{1}^{2} - v_{2}^{2}) = (m/M)(v_{1} - v_{2})^{2}

Take the limit as (m/M) goes to zero.

(v_{1}^{2} - v_{2}^{2}) = 0

So one solution of this is v_{1} = -v_{2}. Another solution is v_{1} = v_{2}, corresponding to the case where the two objects do not collide at all.

One can graph the values of v_{2} and V_{2} against (m/M) and show that as (m/M) goes to zero, the values of the final velocities do indeed smoothly go to the limiting case values. In words, the reason this can happen is that kinetic energy is a scalar, and momentum is a vector, and kinetic energy varies as the square of the speed, while momentum varies as only the first power of speed. Therefore the quantity momentum/energy varies with speed as (1/v). So when v goes to zero, the momentum/energy can be infinite.

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.