1. Today: (Ch. 7)
 Collision
 Center of Mass

2. Changing Mass Example Treat the car as an object whose mass changes
Can be treated as a one-dimensional problem
The car initially moves in the x-direction
The gravel has no initial velocity component in the x-direction
The gravel remains in the car, the total mass of the object is the mass of the car plus the mass of the gravel

3. Changing Mass Example, 2 Treat the problem as a collision
The gravel remains in the car, so it is a completely inelastic collision
The gravel has no momentum in the x-direction before the collision
In both cases,
Momentum is not conserved in the y-direction
There are external forces acting on the car and gravel

4. Problem Solving Strategy – Inelastic Events
Recognize the Principle
The momentum of a system is conserved in a given direction only when the net external force in that direction is zero or negligible
Sketch the Problem
Include a coordinate system
Use the given information to determine the initial and final velocity components
When possible

5. Problem Solving Strategy – Inelastic Events, cont.
Identify the Relationships
Express the conservation of momentum condition for the direction(s) identified
Use the given information to determine the increase or decrease of the kinetic energy
Solve
Solve for the unknown quantities
Generally the final velocity
Check
Consider what the answer means
Does the answer make sense

6. Inelastic Processes and Collisions
Most inelastic processes are similar to collisions
Total momentum is conserved
The separation is just like a collision in reverse

7. Asteroid Splitting Example
Instead of using a rocket to collide with an asteroid, we could try to break it apart
A bomb is used to separate the asteroid into parts
Assuming the masses of the pieces are equal, the parts of the asteroid will move apart with velocities that are equal in magnitude and opposite in direction

8. Center of Mass – Forces
It is important to distinguish between internal and external forces
Internal forces act between the particles of the system
External forces come from outside the system
The total force is the sum of the internal and external forces in the system

9. Forces, cont. The internal forces come in action-reaction pairs
For the entire system, ΣFint = 0
For the entire system, ΣFext = Mtotal aCM
The “cm” stands for center of mass
This is the same form as Newton’s Second Law for a point particle

10. What Is Center of Mass?
The center of mass can be thought of as the balance point of the system
The x- and y-coordinates of the center of mass can be found by
In three dimensions, there would be a similar expression for zCM
To apply the equations, you must first choose a coordinate system with an origin
The values of xCM and yCM refer to that coordinate system

11. Center of Mass, final
All the point particles must be included in the center of mass calculation
This can become complicated
For a symmetric object, the center of mass is the center of symmetry of the object
The center of mass need not be located inside the object

12. Motion of the Center of Mass
The two skaters push off from each other
No friction, so momentum is conserved
The center of mass does not move although the skaters separate
Center of mass motion is caused only by the external forces acting on the system

13. Bouncing Ball and Momentum Conservation
The impulse theorem and the principle of conservation of momentum can be used to treat many different situations
One example is the motion of a pool ball when colliding with the edge of the table
Interested in determining the ball’s velocity after the collision
There is no force acting on the ball in the x-direction
The normal force of the edge of the table exerts an impulse on the ball in the y-direction

14. Pool Ball Example, cont.
Apply conservation of momentum to the x-direction
Solving the resulting equations for the final velocity gives vfy =  viy
Choose the negative
The final velocity is directed opposite to the initial velocity

15. Importance of Conservation Principles
Two conservation principles so far
Conservation of Energy
Conservation of Momentum
Allow us to analyze problems in a very general and powerful way
Example, collisions can be analyzed in terms of conservation principles that completely determine the outcome
Analysis of the interaction forces was not necessary

16. Importance of Conservation Principles
Conservation principles are extremely general statements about the physical world
Conservation principles can be used where Newton’s Laws cannot be used
Careful tests of conservation principles can sometimes lead to new discoveries
Example is the discovery of the neutrino

17. Group Problem Solving COLLECT
A tennis ball launcher projects a ball upward at a 45 degree angle. Air friction acts on the ball, directed against its motion. Enter the proper sign (+, - or 0) for each quantity listed. Explain each answer.
COLLECT
Answer:
On way up:
Weight: -
Friction: -
On way down:
Weight: +
Draw so they can see the arrows

18. Problem Solving Tips
Only use Ef = Eo if the non-conservative forces do no work, otherwise you need to use Wnc = Ef - Eo
Choose a height to be zero. It doesn’t matter what height you choose, as long as you are consistent with that choice through the whole problem. Usually it’s easiest to choose the lowest point in the problem as h=0.
Then, solve the problem!
Really watch the positive and negative work and energy.

19. Example
The Magnum roller coaster at Cedar Point has a vertical drop of 59.4 meters. How fast is the coaster going at the bottom of the hill, if the wheels have frictionless axles?
Assume: v0 is almost zero at top, almost frictionless
Then Ef = Eo
1/2mvf2 + mghf = 1/2mv02 + mgh0
m = ? (ok, will cancel)
vf = ?
hf = 0
vo = 0
h0 = 59.4 m
1/2mvf2 + 0 = 0 + mgh0
vf = sqrt( 2 mfh0/m) = sqrt(2gh0) = 34.1 m/s
or about 76 mi/hr

20. Group Problem Solving
A tennis ball is launched straight upward from ground level, where the potential energy is defined to be zero. Assume that no friction forces act. In the table below, indicate the proper sign (+, -, or 0) and if the quantity is increasing (I), decreasing (D) or not changing (N).
Don’t collect
Answer:
going up:
KE = + D
PE = + I
E =+N
at the top:
KE = 0 I
PE = +D
E = +N
going down:
KE = +I

21. Tomorrow: (Ch. 8)
 Rotational Motion