Collisions.

Example 1. Two balls of sticky putty collide.

Consider two equal mass balls of size m made of sticky putty. One is at rest. One moves toward the other at speed v. They collide and stick together. The combined mass (2m) moves away at speed v'. Draw the picture before and after collision. The conservation of momentum equation is:

[1]

    mv = (2m)v'

From which we conclude that v' = v/2.

So the final speed of the combined putty blob is half the initial speed of the moving ball of putty, v' = v/2. What about the conservation of kinetic energy?

[2]

    (1/2)mv2 = (1/2)(2m)(v')2 + Thermal energy

[3]

    (1/2)mv2 = (1/2)(2m)(v/2)2 + Thermal energy

[4]

    (1/2)mv2 - m(v2/4) = Thermal energy

[5]

    mv2/4 = Thermal energy

So half of the initial kinetic energy goes into thermal energy, heating the putty.

Example 2. Perfectly elastic collision between equal masses.

We now consider a special, but interesting case. Two equal masss balls of mass m collide. One is initially moving with speed v1. The other one is at rest initially. After the collision we cannot claim to know the sizes or directions of the velocities, but we will call them v1' and v2'. Draw the situation before and after collision, assuming that the moving ball is initially to the left of the stationary ball, and the moving ball has speed v1 to the right.

Let's assume that these balls are so elastic that no thermal energy is produced in the collision.

The conservation of momentum and conservation of energy equations may be written.

[7]

    m1v1 = m1v'1 + m2v'2

[8]

    (1/2)m1v12 = (1/2)m1(v'1)2 + (1/2)m2(v'2)2

But in our case m1 = m2, which simplifies these equations to:

[9]

    v1 = v'1 + v'2

[10]

    (v1)2 = (v'1)2 + (v'2)2

Square equation 9.

[11]

    (v1)2 = (v'1)2 + (v'2)2 + 2v1v2

Equation equations 10 and 11 to eliminate v1. The squared terms all drop out.

[12]

    0 = 2v1v2

This has two solutions, since it came from a quadratic equation.

Solution 1: When v'2 = 0, then v'1 = v1. This corresponds to the uninteresting case when the balls actually don't touch, perhaps because the first ball was aimed badly.

Solution 2: When v'1 = 0, then v'2 = v1. This says that the first ball stops dead and the second (initially unmoving) ball moves away with the same speed the first ball had.

This is the result observed in the actual collision between two very elastic balls, say balls made of steel, brass, hard wood, ivory, hard plastic, or even those "high bounce" toy "superballs". And it is the only result that occurs. One might think, if one only considered conservation of momentum, such as result as this might occur: Ball 1, initially moving with speed v might have speed v/4 and ball to have speed 3v/4. That would satisfy conservation of momentum, and many other outcomes could be envisioned that would also satisfy conservation of momentum. But once we insist that we also must satisfy conservation of kinetic energy, then only one result comes from this collision, the one described in Solution 2.

The problem becomes more interesting when we have three elastic balls in a row, initially touching. This becomes the toy called the Newton's cradle. (Actually Newton didn't invent it.) Though the toy is fun to play with, its analysis is beyond the level of this course. Why? Because additional constraints, beyond simply conservation of energy and momentum, apply to the collisions, due to elastic compression, and the velocity of compression pulses through the body of the balls.

Go to the next chapter, Ball bouncing from a massive wall. A surprising and instructive example.
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