**Energy and Elastic Collisions**

You'll recall we had a demonstration of spring constant and ballistic pendulum yesterday.

This was a demonstration of conservation of energy.

The spring seems almost out of sorts, save that we can discuss the energy stored in a spring.

Now we need the potential energy of a spring.

Note it is an energy of position, not motion, thus it is a potential

energy, specifically elastic potential energy.

Again make note of the delta x. This is a change in position from the

equilibrium. It can thus be either a stretching or a compressing of the

spring.

Because the term is squared, the sign turns positive no matter what.

Note it is also a scalar, no direction, and the units are Joules

Finally I ask, can you have a negative potential energy of a spring?

**When a spring is horizontal the only energy**

is kinetic and potential of the spring.

where as when it is vertical the potential energy

can be store both in the spring and due to gravity.

tomorrow there is an example in the workbook (13)

lets solve this problem. With spring constant

k = 250 N/m

m = 5.0 kg

h = 40 cm

is kinetic and potential of the spring.

where as when it is vertical the potential energy

can be store both in the spring and due to gravity.

tomorrow there is an example in the workbook (13)

lets solve this problem. With spring constant

k = 250 N/m

m = 5.0 kg

h = 40 cm

**Take a step back.**

As we do these problems it is all about conservation of energy.

What energy do we have initially, and what energy do we have

finally.

Realize that some problems can be solved with kinematics.

but others not.

As we do these problems it is all about conservation of energy.

What energy do we have initially, and what energy do we have

finally.

Realize that some problems can be solved with kinematics.

but others not.

**Elastic Collisions**

An elastic collision is one where the total

mechanical energy is conserved.

Meaning that the kinetic energy gets transferred

to a spring potential in the atoms, and then that

in turn gets transferred back to kinetic energy,

none of which is lost.

When we had inelastic collisions there were generally

3 unknown variables. Now there are four.

This can be problematic if you don't know either

of the final velocities.

But we now have another equation to help us, energy.

Here are both equations of conservation:

momentum

energy.

If we make some assumptions we can solve for the final velocities

First we will assume the second mass is stationary.

Use this and solve for momentum equation as follows

This we can substitute back into the energy equation.

This solved for one final velocity

now substitute back into find the

other.

Here is that substitution.

Things to note.

This collision is perfectly elastic

It is head on (no glancing only 1D)

Also the second mass is initially at rest

This last seems to be the deal breaker,

but it isn't we can change our coordinate

system to make it work.

Before example consider 3 cases

Ball 1, with a mass of 100 g and traveling at 10 m/s collides head on with ball 2, which has a mass of 300 g and is initially at rest. What is the final velocity of each ball if the collision is

A. perfectly elastic

b. perfectly inelastic?

**As noted we can also do a transfer of**

coordinate system to make this problem

more fun.

coordinate system to make this problem

more fun.

**Energy diagrams**

As noted earlier, one of the tactics to solving

these problems is finding the initial energy, and have that be conserved.

One way of visualizing this is an energy diagram

Most of this book's little diagrams, I don't agree with,

This one however works well.

This first one is of a gravitational potential only

This is a spring potential

This one is even more general, it could be any type of potential

This brings up the idea of Equilibrium positions

An equilibrium is where an object doesn't move.

or where the forces are balanced.

There are multiple types

Stable, and unstable as well as others.

To be stable, a small perturbation from the equilibrium will

cause the object to return to that equilibrium

Unstable is when a small perturbation leads to greater movement

away form the equilibrium

**how many equilibrium**

are there in this graph

A. 1

B. 2

C. 3

D. 4

are there in this graph

A. 1

B. 2

C. 3

D. 4

**How many of these**

equilibrium are

stable?

A. 1

B. 2

C. 3

equilibrium are

stable?

A. 1

B. 2

C. 3

**Lastly the section ends with**

piece on molecular bonds

It uses one of these energy diagrams

and it makes for a bit of an interesting

read, but it glosses over some things.

So we'll leave it until we get to atomic

physics.

piece on molecular bonds

It uses one of these energy diagrams

and it makes for a bit of an interesting

read, but it glosses over some things.

So we'll leave it until we get to atomic

physics.

**A 100 G ball moving to the right at 4.0 m/s catches up and collides with a 400 g ball that is moving to the right at 1.0 m/s. If the collision is perfectly elastic what are the speed and direction of each ball after the collision?**

A 100 g granite cube slides down a 40 degree frictionless ramp. At the bottom, just as it exits onto a horizontal table, it collides with a 200 g steel cube at rest. how high above the table should the granite cube be released to give the steel cube a speed of 150 cm/s. (assume elastic)

**.638 m**

-5 & 5 m/s

2.5 m/s

**-0.8 & 2.2 m/s**

25.8 cm

"So could we ever really have a truly perfectly elastic or inelastic collision?"

"How do Galilean reference frames make mechanical conservation problems easier?"

"I want to get strait all of the equations for all of these different energies. Could you give them to us again?"

"Can you do example 10.6 or 10.7 in class?"

"Can we review how to use reference frames and Galilean transformation?"

"can you explain a little bit more about the energy diagrams?"