1. Chapter 9 Linear Momentum
Demos
Finish up pendulum demo by deriving restoring force.
B. Find cm demo for Virginia.
C. Air track with colliding cars with springs in between them.
Two cars of the same mass with spring between them shows that cm does not move.
Demo comparing an inelastic and elastic collision

2. Assignment 3 Chp 6-2, 6-57, 6-62 9-13, 9-34, 9-54, 9-62, 9-83 Due Monday 9:00 AM

3. Topics Momentum and Its Relation to Force Conservation of Momentum
Collisions
Conservation of Energy and Momentum in Collisions
Elastic & Inelastic Collisions in One Dimension
Elastic & Inelastic Collisions in Two Dimensions

4. Center of Mass (CM)
Center of Mass and Translational Motion
Systems of Variable Mass; Rocket Propulsion

5. Linear Momentum
What is momentum and why is it important? Momentum p is the product of mass and velocity for a particle or system of particles. The product of m and v is conserved in collisions and that is why it is important. It is also a vector which means each component of momentum is conserved. It has units of kg m/s or N-s.

6. Linear Momentum form of Newton’s 2nd Law
Now note we have also a force when the
mass changes

7. Law of Conservation of Linear Momentum

8. Therefore each component of the momentum
Px, Py, Pz is also constant. Gives three equations:
Px= constant
Py = constant
Pz = constant
If one component of the net force is not 0, then that component of momentum is not a constant. For example, consider the motion of a horizontally fired projectile. The y component of P changes while the horizontal component is fixed after the bullet is fired.

9. Conservation of Momentum
Conservation of momentum holds during collisions. A collision takes a short enough time that we can ignore external forces. Since the internal forces are equal and opposite, the total momentum is constant.
Figure 9-4. Caption: Collision of two objects. Their momenta before collision are pA and pB, and after collision are pA’ and pB’. At any moment during the collision each exerts a force on the other of equal magnitude but opposite direction.

10. Types of Collision Two colliding cars.
Rubber ball bouncing off the floor
Two balls colliding on a pool table.
Tennis racquet striking a ball
A bullet striking a target.
High speed electron striking a proton

11. Conservation of Momentum
Example Railroad cars collide and stick: momentum conserved ? Yes.
A 10,000-kg railroad car, A, traveling at a speed of 24.0 m/s strikes an identical car, B, at rest. If the cars lock together as a result of the collision, what is their common speed immediately after the collision?
Figure 9-5.
Solution: Momentum is conserved; after the collision the cars have the same momentum. Therefore their common speed is 12.0 m/s.

12. Explosions Opposite of a collision Chemical or nuclear explosion
Exploding bomb
Firing a bullet
Critical Mass

13. Conservation of Momentum
Example: Rifle recoil.
Calculate the recoil velocity of a 5.0-kg rifle that shoots a kg bullet at a speed of 620 m/s.
Figure 9-7.
Solution: Use conservation of momentum. The recoil velocity of the gun is -2.5 m/s.

14. What happens during a collision on a short time scale?
F(t)
What happens during a collision
on a short time scale?
Consider one object the projectile
and the other the target.

15. J is called the impulse
Change in momentum of the ball.
J is a vector
Also you can “rectangularize” the
graph

16. Shape of two objects while colliding with each other.
Obeys Newtons third Law

17. Racquet and Tennis Ball Collision
Figure 9.8
Racquet and Tennis Ball Collision

18. Example: Andy Rodick has been clocked at serving a
tennis ball up to 149 mph(70 m/s). The time that the
ball is in contact with the racquet is about 4 ms. The
mass of a tennis ball is about 300 grams.
What is the average force exerted by the racquet
on the ball?

19. b) What is the acceleration of the ball?
c) What distance does the racquet go through while
the ball is still in contact?

20. One Dimension Head-on Collision
Total momentum before = Total momentum after
True for any collision

21. One Dimension Elastic Collision
Kinetic energy before=Kinetic energy after
True only for elastic collision

22. This is important to know because you can solve problems
using two equations linear in velocity. This replaces
Kinetic Energy equation.
Giancoli Notation

23. Problem 9-34 Giancoli
A kg ball moving with speed 4.50 m/s has a head–on collision
with a kg ball initially moving in the same direction with speed
3.00 m/s. Determine the final speed and direction of each ball assuming
an elastic collision.

24. Some simple results for a head-on collision with v2i=0

25. You want to understand 3 cases
m1=m2
m2>>m1
m2<<m1

26. Balls bouncing off massive floors, we have m2 >>m1

27. Colliding pool balls The executive toy m2 = m1
Why don’t both balls go to the right each sharing
the momentum and energy?

28. Bowling pin example m2<<m1

30. Types of Collisions
Elastic Collisions: Kinetic energy and momentum are conserved
Inelastic Collision: Only P is conserved.
Kinetic energy is not conserved
Completely inelastic collision. Masses stick together
P is conserved

31. Completely Inelastic Collision
Now look at kinetic energy
Note Ki not equal to Kf

32. Completely Inelastic For equal masses Kf = 1/2 Ki, we lost 50% of Ki
Where did it go? It went into energy of binding the
objects together, such as internal energy, rearrangement
of the atoms, thermal, deformation sound, etc.

33. Collisions in Two Dimensions with particle 2 at rest

34. 2 Dimensional Elastic Collisions with particle 2 at rest
Write down conservation of momentum in x and y directions
separately. Two separate equations because momentum is
a vector. Write down conservation of kinetic energy equation.
3 equations in 2 dimensions

36. Problem 9-88

37. Problem 90-88
A bowling ball travelling at 13.0 m/s
has 5 times the mass of a pin and the
pin goes off at 75 degrees. Find the
speed of the pin, speed of ball, and
the angle of the ball.

38. Improving your Billiards

39. How do you know what angle the billiard ball deflects in an elastic collision?
Head on
Arbitrary Angle
90 Degrees
Line of Action

40. Pool shot
pocket 1 2
Assuming no spin
Assuming elastic collision

41. Where do you aim the Bank shot to make the pocket?
Assuming no english on the balls
Assuming elastic collision and no
bank deformation d

42. How high does a ball bounce up after an almost elastic collision between floor and bouncing ball?
Initial hf
Drop a ball from height h. Its initial energy is all potential, equal to mgh. Just before it collides with the floor, it energy is all kinetic. Just after the bounce, if no friction forces have acted, its energy is the same as before, and is all kinetic, therefore its speed must be the same.
As it rises it converts its kinetic energy to potential energy, doing the reverse of what it did during the fall.
The ball rises to the same height it had initially. This is called an elastic bounce, or elastic collision. In reality no ball is perfectly elastic. Some energy is lost to friction during the collision with the floor when the ball, and also the floor are distorted by the force between them needed to accelerate the ball.
SHOW balls of varying elasticity bouncing.
SHOW double ball bounce. Does this violate energy conservation? What is happening here?
-vi
Before
bounce vf
After
bounce

43. Almost elastic collision
Measuring velocities and heights of balls bouncing from a infinitely massive hard floor
Type of Ball
Coefficient of Restitution
(C.O.R.)
Rebound Energy/
Collision Energy
(R.E./C.E.)
Superball
0.90
0.81
Racquet ball
0.85
0.72
Golf ball
0.82
0.67
Tennis ball
0.75
0.56
Steel ball bearing
0.65
0.42
Baseball
0.55
0.30
Foam rubber ball
0.09
Unhappy ball
0.10
0.01
Beanbag
0.05
0.002
Almost elastic collision
Almost inelastic
collision

44. Two moving colliding objects: HRW Problem 45 ed 6
Show demo comparing the height of a ball bouncing
off the floor compared to one bouncing off another
ball. Explain. m1 m2
DEMO SHOWING SUPERBALL FALLING ON TOP OF BASKETBALL. WHY DOES BALL JUMP SO MUCH HIGHER?
DEMO SHOWING BASEBALL FALLING ON TOP OF BASKETBALL. WHAT MASS IS NEEDED SO THAT BOTH BALLS STOP DEAD?
Just after the little ball
bounced off the big ball
Just after the big ball bounced off the floor
Just before each hits the floor
v1f v m1 -v -v v m2
How high does superball go compared to dropping it off the floor?

45. v1f -v -v v v m1 m2 Just after the big ball bounced off the floor
DEMO SHOWING SUPERBALL FALLING ON TOP OF BASKETBALL. WHY DOES BALL JUMP SO MUCH HIGHER?
DEMO SHOWING BASEBALL FALLING ON TOP OF BASKETBALL. WHAT MASS IS NEEDED SO THAT BOTH BALLS STOP DEAD?
Just after the big ball bounced off the floor
Just after the little ball
bounced off the big ball
Just before each hits the floor
v1f v m1 -v -v v m2

46. -v v For m2 =3m1 v1f = 8/4V =2V superball has twice as much speed.
4 times higher

47. -v v For maximum height consider m2 >>m1 How high does it go?
V1f=3V
V2f = -V
How high does it go?
9 times higher
How about 3 balls?

48. Center of Mass

49. Center of mass (special point in a body)
Why is it important? For any rigid body the motion of the body is given by the motion of the cm and the motion of the body around the cm.
The motion of the cm is as though all of the mass were concentrated there and all external forces were applied there. Hence, the motion is parabolic like a point projectile.
How do you show projectile motion is parabolic?

50. What happens to the ballet dancers head when she raises
her arms at the peak of her jump? Note location of cm relative
To her waist. Her waist is lowered at the peak of the jump.
How can that happen?

51. Center of Mass
As an example find the center of mass of the following system analytically. Note that equilibrium is achieved when the lever arm time the mass on the left equals the similar quantity on the right. If you multiply by g on both sides you have the force times the lever arm, which is called torque. d . m1 m2 x
xcm

52. Center of Mass y x1 m1 m2
xcm x2

53. Problem
2 dimensions
Find xcm and ycm

54. How do you find the center of mass of an arbitrary shape? Show demo

55. How do you find Center of Mass for a system of particles analyticaly

56. Newton’s Second Law for a System of particles: Fnet= Macm
take d/dt on both sides
take d/dt again
Identify ma as the force on each particle

57. Linear Momentum form of Newton’s 2nd Law for a system of particles
Important - a vector quantity that is conserved in interactions.
Now take derivative d/dt of

58. Linear Momentum form of Newton’s 2nd Law for a system of particles
Total momentum is conserved for a system of particles if

59. For a collision the velocity of cm is also constant when no external forces are present
The velocity of the cm is total momentum /total mass.
In general
Consider the total inelastic collision
In the initial state Vcm

61. Did the velocity of the center of mass stay constant for the inelastic collision?
For equal mass objects
After the inelastic collision what is Vcm. Since the particles are stuck together it must be the velocity of the stuck particles
which is v1i/2. Hence, they agree.

62. Problem 9-32 ed 6 HRW
A man of mass w is at rest on a flat car of mass W
moving to the right with speed v0. The man starts running
to the left with speed vrel relative to the flatcar. What is the
change in the velocity of the flatcar Δv = v-v0?
Note W and w are mass

63. Initial momentum = Final momentum
vrel v0 vg v
Initial momentum = Final momentum
Solve for v-v0

64. Problem 9-32 ed 6
vrel v0
What is the change in the velocity of the car Δv =v-v0?
Initial momentum = Final momentum
Basis for the rocket equation

65. Problem The balloon and man is stationary. Vcm=0
The man starts walking up the ladder
with speed vrel relative to the balloon.
In what direction does the balloon move?
What speed does it move at?
What is the speed when the man stops?
How far down did the balloon move?
vrel. m

66. Problem
Balloon moves downward to keep cm fixed
b) What speed does the balloon move at?
c) The balloon stops because his internal motion stops. vB=0
d)Balloon moves= M
uMg
vrel m

67. Systems of variable Mass
Derivation of Rocket Equation

69. Derivation of rocket equation
Both dM/dt and vrel
are negative. This
is like the force
pushing the rocket

70. vrel v

71. When external forces are present
See Giancoli here

72. Example 9-20
A rocket has a mass of kg of which 15000
is fuel. The fuel is spewed out at a rate of 190 kg/s
with a speed of 2800 m/s. If the rocket is launched
vertically upward, find
The thrust
Net force at blast off
Velocity as a function of time
Final velocity at burn out

73. Problem grain B=540 kg/min v =3.20 m/s
What external force is needed to keep the railroad car moving
at constant speed?
The grain that is being added to the car is changing the mass
of the car at the rate of 540 kg/min. There is friction between
The grain and cart.

74. Problem grain B=+540 kg/min v =3.20 m/sx
vrel is the relative velocity of the grain relative to the cart
in the x direction. vrel = 0 - v
dm/dt is the rate at which mass is changing

75. A rope of length L lies in a straight line on a frictionless table, except for a very small piece at one end which hangs down through a hole in the table. This piece is released, and the rope slides down through the hole. What is the speed of the rope at the instant it loses contact with the table?
Use energy conservation and concept of cm L
L/2

76. 2nd Solution

77. (b) A rope of length L lies in a heap on a table, except for a very small piece at one end which hangs down through a hole in the table. This piece is released, and the rope unravels and slides down through the hole. What is the speed of the rope at the instant it loses contact with the table? (Assume that the rope is greased, so that it has no friction with itself.) x L