General Principle

What are the laws that govern the distribution of motion among colliding bodies, whether elastic or inelastic [static or energetic]?

We will derive these laws from only one principle. From the same principle, we will derive the laws of mechanical equilibrium.

General Principle

When a change occurs in Nature, the quantity of action necessary for that change is as small as possible.

The quantity of action is the product of the mass of the bodies times their speed and the distance they travel.

When a body is transported from one place to another, the action is proportional to:

• the mass of the body
• its speed
• the distance it traveled

Problem 1: Laws of Motion for Inelastic Bodies

Two inelastic bodies of masses m1 and m2 and moving in the same direction with speeds v1 and v2 respectively.

Body 1 moves faster. It overtakes Body 2 and collides with it.

After the collision, let the common velocity of the two bodies is vf leading to v2 < vf < v1 [i.e. the resulting speed of the 2 bodies after collision is in between the fastest and the slowest body].

Background Versus Foreground

The change in the universe is that, Body 1 was moving at a speed v1 and was covering a distance v1 per unit time.

But now it moves only at a speed vf and covers only a distance vf per unit time.

Whereas Body 2 was moving only at a speed v2 and was covering only a distance v2 per unit time.

But now it moves at speed vf and covers a distance vf 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 - vf and covering a distance v1 - vf per unit time.
• 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 vf - v2 and covering a distance vf - v2 per unit time.

The motion of these immaterial planes 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 - vf)^2 and m2(vf - v2)^2 , the sum of which should be minimized.

Thus, we have

..

From this we can derive the final speed:

..

In this case, the 2 bodies are moving in the same direction.

• The quantity of momentum produced and destroyed is the same.
• The total momentum is constant, being the same after the impact as beforehand.

If 2 bodies are moving towards each other, the second speed v2 is made negative.

In that case, the final speed is

…

If the second body is at rest before the impact, v2 = 0 and the final speed is

…

If the first body encounters an impassable barrier, one can consider that barrier as a body of infinite mass at rest. Since is m2 infinite, the final speed vf = 0

..

What about perfectly elastic bodies?