MOMENTUM, IMPULSE, AND COLLISIONS

Momentum and Its Relation to Force

Momentum is a vector symbolized by the symbol p , and is defined as Screen Shot 2021-02-25 at 15.44.22

The rate of change of momentum is equal to the net force: Screen Shot 2021-02-25 at 15.44.28

Conservation of Momentum

During a collision, measurements show that the total momentum does not change:

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Conservation of momentum can also be derived from Newton’s laws. A collision takes a short enough time that we can ignore external forces. Since the internal forces are equal and opposite, the total momentum is constant.
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For more than two objects Screen Shot 2021-02-25 at 15.48.50
Or, since the internal forces cancel
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Collisions and Impulse

During a collision, objects are deformed due to the large forces involved.

This quantity is defined as the impulse, J: Screen Shot 2021-02-25 at 15.57.54
The impulse is equal to the change in momentum:
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Since the time of the collision is often very short, we may be able to use the average force, which would produce the same impulse over the same time interval.

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Conservation of Energy and Momentum
in Collisions

Momentum is conserved in all collisions. Collisions in which kinetic energy is conserved as well are called elastic collisions, and those in which it is not are called inelastic.

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Elastic Collisions in One Dimension

Here we have two objects colliding elastically. We know the masses and the initial speeds.
Since both momentum and kinetic energy are conserved, we can write two equations. This allows us to solve for the two unknown final speeds. Screen Shot 2021-02-25 at 16.27.35

Inelastic Collisions

With inelastic collisions, some of the initial kinetic energy is lost to thermal or potential energy. Kinetic energy may also be gained during explosions, as there is the addition of chemical or nuclear energy.
A completely inelastic collision is one in which the objects stick together afterward, so there is only one final velocity.

Collisions in Two or Three Dimensions

Conservation of energy and momentum can also be used to analyze collisions in two or three dimensions, but unless the situation is very simple, the math quickly becomes unwieldy.

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Problem solving:

  1. Choose the system. If it is complex, subsystems may be chosen where one or more conservation laws apply.
  1. Is there an external force? If so, is the collision time short enough that you can ignore it?
  1. Draw diagrams of the initial and final situations, with momentum vectors labeled.
  1. Choose a coordinate system.

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  1. Apply momentum conservation; there will be one equation for each dimension.
  1. If the collision is elastic, apply conservation of kinetic energy as well.
  1. Solve.
  1. Check units and magnitudes of result.

Center of Mass (CM)

In (a), the diver’s motion is pure translation; in (b)
it is translation plus rotation.

There is one point that moves in the same path a Particle would take if subjected to the same force as the diver. This point is called the center of mass (CM).

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The general motion of an object can be considered as the sum of the translational motion of the CM, plus rotational, vibrational, or other forms of motion about the CM.

The general motion of an object can be considered as the sum of the translational motion of the CM, plus rotational, vibrational, or other forms of motion about the CM.

For two particles, the center of mass lies closer to the one with the most mass: Screen Shot 2021-02-25 at 16.40.24 Screen Shot 2021-02-25 at 16.40.28

The center of gravity is the point at which the gravitational force can be considered to act. It is the same as the center of mass as long as the gravitational force does not vary among different parts of the object. Screen Shot 2021-02-25 at 16.41.06

The center of gravity is the point at which the gravitational force can be considered to act. It is the same as the center of mass as long as the gravitational force does not vary among different parts of the object. Screen Shot 2021-02-25 at 16.41.48

Center of Mass and Translational Motion

The total momentum of a system of particles is equal to the product of the total mass and the velocity of the center of mass.
The sum of all the forces acting on a system is equal to the total mass of the system multiplied by the acceleration of the center of mass: Screen Shot 2021-02-25 at 16.42.57
Therefore, the center of mass of a system of particles (or objects) with total mass M moves like a single particle of mass M acted upon by the same net external force.

Systems of Variable Mass; Rocket Propulsion

Applying Newton’s second law to the system shown gives: Screen Shot 2021-02-25 at 16.44.01 Screen Shot 2021-02-25 at 16.44.05 Screen Shot 2021-02-25 at 16.44.10

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