Forest UCM EnergyIntPart

Energy of Interacting particles

Translational invariance

Consider two particles that interact via a conservative force

Let identify the location of object 1 from an arbitrary reference point and locate the second object.

The vector that points from object 2 to object 1 may be written as

The distance between the two object is given as the length of the above vector

If the force is a central force

Notice
The interparticle force is independent of the coordinate system's position, only the difference betweenthe positions matters

If object 2 was fixed so it is not accelerating and we place the origin of the coordinate system on object 2

Then the force is that of a single object

One potential for Both Particles

Both forces from same potential

If the above force is conservative then a potential exists such that

Newton's 3rd law requries that

or

You can find the net external force on a body in the system once you have the potential for the system

Total work given by one potential

The total work is the sum of

the work done by as object 1 moves through

plus

the work done by as object 1 moves through

This NET work can be determine by taking the derivative of the potential energy

Proof:

Total Mechanical Energy conservation

The work done by as object 1 moves through is given by the work energy theorem as

similarly for

Thus

or

constant

Elastic Collisions

Definition

BOTH Momentum and Energy are conserved in an elastic collision

Example

Consider two object that collide elastically

Conservation of Momentum
Conservation of Energy

When the initial and final states are far away fromthe collision point

arbitrary constant

Example

Consider an elastic collision between two equal mass objecs one of which is at rest.

Conservation of momentum
Conservation of Energy

Square the conservation of momentum equation

compare the above conservation of momentum equation with the conservation of energy equation

and you conclude that