1. Review Chap. 8 Momentum, Impulse, and Collisions
PHYS 218 sec
Review
Chap. 8
Momentum, Impulse, and Collisions

2. What you have to know Momentum and impulse Momentum conservation
Various collision problems
The typical examples where momentum conservation is used are collisions!
Center of mass
Section 8.6 is not in the curriculum.

3. Newton’s 2nd law of motion
This cannot be applied to the problems where mass m is changing.
E.g.
Rocket propulsion problem. (See Sec 9.6)
Original form of the 2nd law of motion written by Newton
Momentum is a vector quantity.

4. Impulse-momentum theorem
The impulse-momentum theorem
Impulse
Impulse is a vector quantity.
Impulse-momentum theorem
The change in momentum of a particle during a time interval equals the impulse of the net force.

5. Impulse-momentum theorem
General definition
Impulse-momentum theorem
Impulse: area under the curve of net force in F-t graph
Large force for a short time
Smalle force for a longer time
Same area
Same impulse

6. identical Momentum vs. Kinetic Energy
Integral form of Newton’s 2nd law
identical
Differential form of Newton’s 2nd law

7. Ex 8.2
A ball hits a wall
before
after

8. First decompose the momentum into x and y directions
Ex 8.3
This is a 2-dim. motion.
First decompose the momentum into x and y directions
After
Before

9. Conservation of Momentum External force
Internal forces
Isolated system: External force = 0
Newton’s 2nd law
Newton’s 3rd law
You can add these forces to obtain the net (internal) force for the system
The sum of these forces cannot be applied to a single object.

10. Conservation of Momentum
Total momentum of an isolated system is conserved.
Total momentum
Momentum conservation gives a vector equation.

11. Note that momentum conservation is valid for an isolated system.
Ex 8.4
Recoil of a rifle
Rifle + bullet
Before
Note that momentum conservation is valid for an isolated system.
So it is useful when the system changes. Therefore, draw diagrams for “before” & “after” the event
After
Rifle
bullet
The negative sign means that the rifle is moving back!
change the unit so that the both masses have the same unit

12. The sum of these two momenta vanishes as it should be.
Ex 8.4 (cont’d)
The sum of these two momenta vanishes as it should be.
The kinetic energy is NOT conserved.
The kinetic energy before the event was 0.

13. Collision along a straight line
Ex 8.5
Collision along a straight line
Before A B
After
If you have a collision problem, always use momentum conservation. A B

14. Collision in a horizontal plane
Ex 8.6
Collision in a horizontal plane
Before
After
Again, we use momentum conservation.
Also note that momentum is a vector quantity.
Use the numbers above

15. Momentum conservation and collisions
The key to solve collision problems is Momentum conservation.
At the instant of collision, very complex and very large forces are acting on the bodies.
Then the other forces can be neglected and the system is almost an isolated system. g momentum conservation can be applied
The forces are very complex and we do not know much about the forces. g momentum conservation can give useful information for the motions of the colliding bodies
Elastic collision
If the forces between the colliding bodies are conservative, no mechanical energy is lost The total kinetic energy of the system is conserved.
For elastic collisions, the total momentum and the total kinetic energy of the system are conserved.

16. Completely inelastic collision
In this collision, the total kinetic energy of the system after the collision is less than before the collision.
The forces during the collision are non-conservative.
Completely inelastic collision
This is a special case of inelastic collisions.
This is where the colliding bodies stick together and move as one body after the collision.

17. Completely inelastic collisions
What happens to momentum and kinetic energy in a completely inelastic collision of two bodies
Before A B
After
Consider a 1-dim collision where vB1 = 0 A B

18. (A) (B) Ex 8.8 The ballistic pendulum
This is a device which can measure the speed of a bullet.
(A)
(B)
Before collision
Immediately after collision
Swing of the body after collision
Collision process
One-body motion
This gives a relation between v1 and v2.
This gives a relation between v2 and the height y.
So, if you know y, you can know v2 and v1
Analyze this event in two stages !

19. By measuring the height y, you can know the speed of the bullet
Ex 8.8 (cont’d)
The first stage: (A) collision
The second stage: (B) one-body motion
Combining the two
By measuring the height y, you can know the speed of the bullet

20. Ex 8.9
Automobile collision
Completely inelastic collision
After
Before

21. Consider 1-dim elastic collision.
Elastic collisions
Consider 1-dim elastic collision.
Before A B
After A B
These give TWO equations.
You now have TWO unknowns (the final velocities).
Therefore, you can completely determine the unknowns.

22. Elastic collisions: one body initially at rest
rearrange
rearrange
Before A B
After A B

23. The motion of the heavy object doesn’t change.
Before A
Before B B A
After
After A B B A
The motion of the heavy object doesn’t change.
The light object moves in the direction of the heavy one but (two times) faster than the heavy one.
The motion of the heavy object doesn’t change.
The light object moves to the opposite direction but with same speed.

24. Elastic collisions & relative velocity
The velocity of B relative to A after the collision
The negative of the velocity of B relative to A before the collision
If you watch the motion of A sitting on B, then you will see:
A is approaching to B with speed v before the collision.
After the collision, A is moving away from B with the same speed

25. Two-dim elastic collision
(If this is a head-on collision, it is the same as the 1-dim collision.)
Ex 8.12
Before
FOUR unknowns after collision
(two velocity vectors)
After
Momentum conservation gives two eqs.
Kinetic energy conservation gives one eq.
So totally THREE equations.
To uniquely determine the final velocities, we need ONE more information for the final velocities

26. Ex 8.12 (cont’d)

27. We now introduce the concept of center of mass (CM)
i-th particle

28. Motion of the center of mass
Total momentum of the system
Momentum of the CM
You can represent an extended object as a point-like particle; the total mass of the object is at its Center of Mass (CM)

29. This is the trajectory of the shell if it doesn’t explode.
When a shell explodes `
Motion of the CM
This is the trajectory of the shell if it doesn’t explode.