Lesson 15 Conservation of Momentum

The momentum of an object is m x v; when no external forces act on a system the total momentum is constant, this is known as the conservation of momentum. The total quantity of motion in the universe is fixed. In other words, if a body is not interferred with it will move at a constant speed in a straight line (Inertia). The body does not come to rest but transfers motions to other bodies; the total energy is conserved. The change in motion is proportional to the force impressed, and is made in the same direction of the straight line in which the force is impressed.

Motion = Momentum

p = m v

Momentum = Mass x Velocity

All of Newton's laws can be represented by differential equations:

F = dp/dt (second law)

A single particle of some mass has p = m v

F = d   p
       dt

No force 0 = d   p
                     dt
p is constant and m v is constant as the motion of the object is constant

Billiards is a game where the conservation of momentum heavily applies. For instance, if a ball is traveling in a straight line (second law, inertia) it has momentum. If two billiard balls come in contact with each other, each ball applies a momentary force causing momentum to change. According to Newton's third law, this change is equal and opposite.

F1 = -F2
dp1/dt = -dp2/dt
d/dt(p1 + p2) = 0 because the total momentum of two balls taken together does not change and is constant.

This rule applies to any number of balls and also to the atoms of which the balls are made. Momentum is always conserved. Each ball is composed of atoms and smaller parts supplying equal and opposite forces to each other. Each ball behaves as if it were a single body with all its mass concentrated at a single point. This is known as the center of mass, where focus must be drawn when calculating velocity and acceleration of a compound body when no net outside force acts on a compund system. No matter what happens to its individual parts the center of mass continues to move at a constant speed in a straight line. Recoil, as when shooting a shotgun, is a prime example of conservation of momentum.



Below is a video that shows how the center of mass moves at a constant rate with the conditions stated above:


The conservation of energy has many forms, as enrgy is conserved strictly and absolutely no matter the masses of the bodies, or type of energy.

The sum of the momentum of all bodies is a conserved quantity that is always the same even with two or more bodies. When there is no outside force, the center of mass moves at a constant velocity. For instance, a planet pulls a moon just as the moon pulls the planet with equal and opposite force.


F1 + F2 = 0, when the derivative of something is zero, the original value was constant.

p = m v

v = p/m

Kinetic energy: k = 1/2 m v ^2
                          k = 1/2 m (p/m x p/m)
                          k = p^2/2m

One ball strikes another, this reveals that at rest, there are only two possible outcomes:

1.) p0 = p1 + p2

This is a vector equation that form a triangle. In the collision of billiard balls, only kinetic energy is observed (ignore heat from collison), and kinetic energy is also conserved.

k0 = k1 + k2           k = p^2/2m
p(0)^2/2m = p(1)^2/2m + p(2)^2/2m

p(0)^2 = p(1)^2 + p(2)^2 (The pythagorean theorem)

p(1) is perpendicular to p(2), so when one ball strikes another, the balls come off at right angles.

p(0) = p(1) + p(2)             p(0)^2 = p(1)^2 + p(2)^2

2.)  p(0) = p(2)                  p(0)^2 = p(2)^2

The ball transfers its momentum to the other ball and stops.




Physicists use collisions as the sole method for studying the subatomic world.

Descartes chose to look at nature in terms of mathematics and pictured philosophy as a tree. Where the base of the tree is "physics" with all other sciences branching from it.

Below is a video that explains the conservation of momentum in mathematical terms: