1. Momentum Conservation: Review
The concept of momentum conservation is one of the most fundamental principles in physics.
This is a component (vector) equation.
We can apply it to any direction in which there is no external force applied.
You will see that we often have momentum conservation even when kinetic energy is not conserved.

2. Comment on Energy Conservation
We have seen that the total kinetic energy of a system undergoing an inelastic collision is not conserved.
Mechanical Energy is lost:
Heat (bomb)
Bending of metal (crashing cars)
Kinetic energy is not conserved since dissipative work is done during an inelastic collision! (here, KE equals mechanical energy)

3. Comment on Momentum Conservation
Total momentum, PT, along a certain direction is conserved when there are no external forces acting in this direction.
F = ma = PT/Δt says this has to be true!! (Newton’s Laws)
In general, momentum conservation is easier to satisfy than mechanical energy conservation.
Remember: in the absence of external forces, total energy (including heat…) of a system is always conserved even when mechanical energy is not conserved.
How much do two objects that inelastically collide heat up?

4. Collisions
A box sliding on a frictionless surface collides and sticks to a second identical box which is initially at rest.
What is the ratio of initial to final kinetic energy of the system?
(a) 1
(b)
(c) 2

5. Solution No external forces in the x direction, so PX is constant. v m

6. Solution
Compute kinetic energies: v m m m m
v / 2

7. Another solution We can write
P is the same before and after the collision.
The mass of the moving object has doubled, hence the kinetic energy must be half. m m m m

8. Elastic Collisions
Elastic means that kinetic energy is conserved as well as momentum.
This gives us more constraints
Billiards (2-D collision)
The colliding objects have separate motions after the collision as well as before.
Final
Initial

9. Elastic Collision in 1-D what has to happen
Why is this elastic? m2 m1
initial
v1,i
v2,i
Maybe, it depends… x
v1,f
v2,f
final m1 m2
Kinetic energy  potential energy  kinetic energy
The spring is conservative

10. Elastic Collision in 1-D the spring is conservative
v1,i
v2,i
before m1 m2
Conserve PX: (no external forces!)
m1v1,i + m2v2,i = m1v1,f + m2v2,f x
v1,f
v2,f
after
Conserve Kinetic Energy: (it’s elastic!)
1/2 m1v21,i + 1/2 m2v22,i = 1/2 m1v21,f + 1/2 m2v22,f
Suppose we know v1,i and v2,i
We need to solve for v1,f and v2,f
Should be no problem equations & 2 unknowns!

11. Elastic Collision in 1-D
However, solving this can sometimes get a little bit tedious since it involves a quadratic equation!!
m1v1,i + m2v2,i = m1v1,f + m2v2,f
1/2 m1v21,i + 1/2 m2v22,i = 1/2 m1v21,f + 1/2 m2v22,f
momentum
energy

12. Elastic Collision in 1-D special case: equal masses
If the masses of the two objects are equal the algebra is not too bad. Let’s see what we get…
Divide through by m = m1 = m2
m1v1,i + m2v2,i = m1v1,f + m2v2,f
1/2 m1v21,i + 1/2 m2v22,i = 1/2 m1v21,f + 1/2 m2v22,f
momentum
energy
v1,i + v2,i = v1,f + v2,f
v21,i + v22,i = v21,f + v22,f

13. Elastic Collision in 1-D special case: equal masses
Now just rearrange equations to bring v1,i and v1,f to left hand side, and v2,i and v2,f to rhs
Divide through energy equation by momentum equation which gives
Particles just trade velocities in 1-D elastic collision of equal mass objects
v1,i - v1,f = v2,f - v2,i
momentum
energy
v21,i - v21,f = v22,f - v22,i
(v1,i - v1,f )(v1,i + v1,f ) = (v2,f - v2,i )(v2,f + v2,i )
v1,i + v1,f = v2,f + v2,i
v2,f = v1,i
v1,f = v2,i
v1,i - v1,f = v2,f - v2,i

14. Elastic Collision in 1-D general case: unequal masses
Conserve linear momentum and mechanical energy, but now the masses are different:
Divide through energy equation by momentum equation which gives
Now solving these two linear equations is only a bit more complicated
m1(v1,i - v1,f ) = m2(v2,f - v2,I )
momentum
energy
m1(v21,i - v21,f ) = m2(v22,f - v22,I )
m1(v1,i - v1,f )(v1,i + v1,f ) =m2 (v2,f - v2,i )(v2,f + v2,i )
v1,i + v1,f = v2,f + v2,i
m1(v1,i - v1,f ) = m2(v2,f - v2,I )

15. Elastic Collision in 1-D general case: unequal masses
Algebra just gave us the following equations based on conservation of momentum and mechanical energy:
Now just solve for final velocities, v1,f and v2,f in terms of v1,i and v2,i
v1,i + v1,f = v2,f + v2,i
m1(v1,i - v1,f ) = m2(v2,f - v2,I )
v2,f = v1,i
v1,f = v2,i
When m1 = m2

16. Recap of lecture
Elastic Collision –a collision in which the total kinetic energy after the collision equals the total kinetic energy before the collision.
If m2 is initially at rest, then you can use: conservation of energy and conservation of momentum, or:

17. Example
A ball with mass 2.5x10-2 kg moving at 2.3 m/s collides with a stationary 2.0x10-2 kg ball. If the collision is elastic, determine the velocity of each ball after the collision.

18. Solution
A ball with mass 2.5x10-2 kg moving at 2.3 m/s collides with a stationary 2.0x10-2 kg ball. If the collision is elastic, determine the velocity of each ball after the collision.
K.E.
Momentum
Given
Re-arranging the Momentum equation with v2i=0, we obtain:

19. Substitute this into the K.E. equation:
Substitute in: