Linear Momentum

We will now begin to study momentum conservation in order to best establish an experimental link to Newton's Laws.

What is Momentum? Momentum is the tendency for an object in motion to stay in in motion.

We have already observed this behavior in the case of motion in one dimension without friction. Once an impulse is given to an object, it just keeps moving in that same direction.

If there are no forces acting on an object an object remains at constant velocity. This is Newton's First Law: The Law of Inertia

A more physical way to state Newton's First Law is to say that the momentum of an object remains constant unless a force acts on that object. This implies that a force produces a change in the momentum of an object. This is Newton's Second Law

Again, we have already explored Newton's Second Law when we did experiments with impulse. Large impulses produced a larger velocity.

Linear momentum is defined as:

p = mv

Since velocity is part of momentum, then momentum has a direction and is thus a vector quantity.

Conservation of Momentum is a rule of mechanics. your intution has already told you about it.

Consider hitting a baseball. The bat, with some mv, makes impulsive contact with a ball - m_bat is greater than m_ball and since:

(mv)_bat = (mv)_ball

then v_ball is greater than v_bat (provided that you hit the ball and not just air).

Remember that impulse x time = change in momentum.

So an impulsive force (force applied over a small time dt) is a change in momentum:

F * dt = m * dv

where dv = a change in velocity (the force produce an acceleration).

Same principle holds in the situation of cannon recoil:

Applications of Conservation of Momentum:

• Elastic Collisions: Collisions that conserve both momentum and energy. Most collisions in the macroscopic world do not do this.

  For instance, a collision on a football field does not conserve energy. Most of the energy is dissipated in the form of sound waves and shock stress to the participants.

  In the atomic world, collsions between atoms are often elastic.

• Inelastic Collisions: Collisions that conserve momentum but not energy. The total amount of Energy which is lost, however, is dictated exactly by momentum conservation. That is, the energy which is lost equals that which is required to maintain momentum conservation.

  Suppose we have two identical mass objects, one moves a velocity of +v and one moves at velocity of -v. These two objects will collide. We observed that after the collision both objects were at rest. Why?

  Conservation of momentum is a relatively straightforward concept if you simply calculate the momentum of the system before and after the collision.

  In this case the momentum before the collision was +mv (for object 1) and -mv (for object two). Thus the total momentum of the system (the system being just these two objects) is zero. Whatever happens to this system, this total momentum must be conserved. Hence, when the objects collide we immediately see the manifestation of momentum conservation both objects stop.

  What about energy conservation? Well the kinetic energy of both objects was the same (1/2 mv²) initially and so the total kinetic energy of the system is mv². After the collision, v is zero and hence the kinetic energy of the system is zero Energy was not conserved.