Problem 2: Laws of Motion for Elastic Bodies

Let there be two elastic bodies of masses m1 and m2 moving in the same direction with speeds v1 and v2, respectively.

v1 is faster so that it overtakes the m2 and collides with it.

Let u1 and u2 represent the speeds of the 2 bodies after the collision.

The sum or difference of these speeds is the same as that before the collision.

m1 was moving at speed v1 and was covering a distance v1 per unit time. The universe changed so that now m1 moves at speed u1 and covers a distance u1 per unit time.

m2 was moving at speed v2 covering a distance v2 per unit time. Now it moves at speed u2 and covers a distance u2 per unit time.

This change would be the same if:

• while Body 1 was moving at speed v1 and was covering a distance v1 per unit time, it was being transported backwards by an invisible, massless plane moving at speed v1 - u1 and covering a distance v1 - u1 per unit time
• while Body 2 was moving at speed v2 and was covering a distance v2 per unit time, it was being transported forwards by an invisible, massless plane moving at speed u2 - v2, covering a distance u2 - v2 per unit time.

The motion of these immaterial planes conveying the masses m1 and m2 are the same, regardless of whether the masses are moving relative to these planes or are at rest.

Hence, the quantities of action produced in Nature are m1(v1 - u1)^2 and m2(u2 - v2)^2, the sum of which should be minimized.

Thus, we have:

…

For elastic bodies, the relative speed after the impact should equal the relative speed before the impact.

Hence, we have u2 - u1 = v1 - v2, as u2 = u1 + v1 - v2 and, thus, dv2 = dv1.

Substitution into the preceding equation yields the final speeds:

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and

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When 2 bodies are moving towards each other, we make the second speed negative. In that case, the final speeds are:

…

and

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If the second body is at rest before the impact, then v2 = 0, then the final speeds are:

…

and

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If the first body encounters an impassable barrier, one can consider that barrier as a body of infinite mass at rest.

In that case, the final speed u1 = -v1 i.e., the first mass rebounds at the same speed with which it struck the barrier.

If one takes the sum of the kinetic energies, one sees that they are the same after the impact as before; thus:

…

the sum of the kinetic energies is conserved after the impact. However, this conservation applied only to elastic bodies, and not to inelastic bodies.

The general principle that applies to both types of bodies is that the quantity of action required to cause a change in Nature is as small as possible.

This principle is so universal and so fruitful that one can also derive the law of mechanical equilibrium from it.

At equilibrium, there is no difference between elastic and inelastic bodies.*