1. Momentum, Energy, and Collisions
Lecture 12
Momentum, Energy, and Collisions

2. Announcements Assignment 6 due Wednesday, Oct 5 (11:59pm)
EXAM: October 13 (through Chapter 9)
Look for messages regarding special TA office hours
Reminder that formula sheet for Test 2 now on Collab under Midterms
Practice problems will be attached to posted slides (this afternoon, I promise)
Practice test from last year on Collab under Resources

3. Reading and Review

4. Linear Momentum
Impulse
With no net external force:

5. A plate drops onto a smooth floor and shatters into three pieces of equal mass. Two of the pieces go off with equal speeds v along the floor, but at right angles to one another. Find the speed and direction of the third piece.

6. We know that px=0, py = 0 in initial state
A plate drops onto a smooth floor and shatters into three pieces of equal mass. Two of the pieces go off with equal speeds v along the floor, but at right angles to one another. Find the speed and direction of the third piece.
We know that px=0, py = 0 in initial state
and no external forces act in the horizontal 

7. An 85-kg lumberjack stands at one end of a 380-kg floating log, as shown in the figure. Both the log and the lumberjack are at rest initially.
(a) If the lumberjack now trots toward the other end of the log with a speed of 2.7 m/s relative to the log, what is the lumberjack’s speed relative to the shore? Ignore friction between the log and the water.
(b) If the mass of the log had been greater, would the lumberjack’s speed relative to the shore be greater than, less than, or the same as in part (a)? Explain.
(c) Check (b) by assuming log has mass of 450 kg.

9. Rolling in the Rain
An open cart rolls along a frictionless track while it is raining. As it rolls, what happens to the speed of the cart as the rain collects in it? (Assume that the rain falls vertically into the box.)
a) speeds up
b) maintains constant speed
c) slows down
d) stops immediately
Answer: c

10. Rolling in the Rain
An open cart rolls along a frictionless track while it is raining. As it rolls, what happens to the speed of the cart as the rain collects in it? (Assume that the rain falls vertically into the box.)
a) speeds up
b) maintains constant speed
c) slows down
d) stops immediately
Because the rain falls in vertically, it adds no momentum to the box, thus the box’s momentum is conserved. However, because the mass of the box slowly increases with the added rain, its velocity has to decrease.

11. Gun Control
When a bullet is fired from a gun, the bullet and the gun have equal and opposite momenta. If this is true, then why is the bullet deadly (whereas it is safe to hold the gun while it is fired)?
a) it is much sharper than the gun
b) it is smaller and can penetrate your body
c) it has more kinetic energy than the gun
d) it goes a longer distance and gains speed
e) it has more momentum than the gun
Answer: c

12. Gun Control
When a bullet is fired from a gun, the bullet and the gun have equal and opposite momenta. If this is true, then why is the bullet deadly (whereas it is safe to hold the gun while it is fired)?
a) it is much sharper than the gun
b) it is smaller and can penetrate your body
c) it has more kinetic energy than the gun
d) it goes a longer distance and gains speed
e) it has more momentum than the gun
Even though it is true that the magnitudes of the momenta of the gun and the bullet are equal, the bullet is less massive and so it has a much higher velocity. Because KE is related to v2, the bullet has considerably more KE and therefore can do more damage on impact.

13. Two objects collide... and stick
mass m
No external forces... so momentum of system is conserved
initial
px = mv0
mv0 = (2m)vf
vf = v0 / 2
final
px = (2m)vf
A completely inelastic collision: no “bounce back”

14. What about energy? Kinetic energy is lost! vf = v0 / 2 initial final
mass m
vf = v0 / 2
initial
final
Kinetic energy is lost!
KEfinal = 1/2 KEinitial

15. Inelastic Collisions This is an example of an “inelastic collision”
Collision: two objects striking one another
“Elastic” collision <=> “things bounce back”
Completely inelastic collision: objects stick together afterwards... no thing “bounces back”
Time of collision is short enough that external forces may be ignored so momentum is conserved
Inelastic collision: momentum is conserved but kinetic energy is not

16. Elastic vs. Inelastic Completely inelastic collision:
colliding objects stick together, maximal loss of kinetic energy
Inelastic collision:
momentum is conserved but kinetic energy is not
Elastic collision:
momentum and kinetic energy is conserved.

17. Completely Inelastic Collisions in One Dimension
Solving for the final momentum in terms of initial velocities and masses, for a 1-dimensional, completely inelastic collision between unequal masses:
Completely inelastic only
(objects stick together, so have same final velocity)
Momentum Conservation:
KEfinal < KEinitial

18. Ballistic pendulum: the height h can be found using conservation of mechanical energy after the object is embedded in the block.
vf = m v0 / (m+M)
momentum conservation
in inelastic collision
KE = 1/2 (mv0)2 / (m+M)
energy conservation
afterwards
PE = (m+M) g h
hmax = (mv0)2 / [2 g (m+M)2]

19. Binomial expansion
If  <<1 then (1+ )n ~1+n  Ex.

20. Velocity of the ballistic pendulum
Pellet Mass (m): x10-3 kg
Pendulum Mass (M): kg
Wire length (L): m

21. Crash Cars I
a) I
b) II
c) I and II
d) II and III
e) all three
If all three collisions below are totally inelastic, which one(s) will bring the car on the left to a complete halt?
Answer: e

22. Crash Cars I
a) I
b) II
c) I and II
d) II and III
e) all three
If all three collisions below are totally inelastic, which one(s) will bring the car on the left to a complete halt?
In case I, the solid wall clearly stops the car.
In cases II and III, because ptot = 0 before the collision, then ptot must also be zero after the collision, which means that the car comes to a halt in all three cases.

23. Crash Cars II
a) I
b) II
c) III
d) II and III
e) all three
If all three collisions below are totally inelastic, which one(s) will cause the most damage (in terms of lost energy)?
Answer: c

24. Crash Cars II
a) I
b) II
c) III
d) II and III
e) all three
If all three collisions below are totally inelastic, which one(s) will cause the most damage (in terms of lost energy)?
The car on the left loses the same KE in all three cases, but in case III, the car on the right loses the most KE because KE = mv2 and the car in case III has the largest velocity.

25. Inelastic Collisions in 2 Dimensions
For collisions in two dimensions, conservation of momentum is applied separately along each axis:
Energy is not a vector equation: there is only 1 conservation of energy equation
Momentum is a vector equation: there is 1 conservation of momentum equation per dimension

26. In elastic collisions, both kinetic energy and momentum are conserved.
One-dimensional elastic collision:

27. Elastic Collisions in 1-dimension
We have two equations:
conservation of momentum
conservation of energy
and two unknowns (the final speeds).
For given case of v2i = 0, solving for the final speeds:
Note: relative speed is conserved for head-on (1-D) elastic collision

28. Limiting cases

29. Limiting cases

30. Limiting cases

31. Toy Pendulum Incompatible!
Could two balls recoil and conserve both momentum and energy?
Incompatible!

32. Elastic Collisions I 2 1 at rest v a) situation 1 b) situation 2
Consider two elastic collisions: 1) a golf ball with speed v hits a stationary bowling ball head-on. 2) a bowling ball with speed v hits a stationary golf ball head-on. In which case does the golf ball have the greater speed after the collision?
a) situation 1
b) situation 2
c) both the same
Answer: b v 2 1
at rest

33. Elastic Collisions I v 1 v 2 a) situation 1 b) situation 2
Consider two elastic collisions: 1) a golf ball with speed v hits a stationary bowling ball head-on. 2) a bowling ball with speed v hits a stationary golf ball head-on. In which case does the golf ball have the greater speed after the collision?
a) situation 1
b) situation 2
c) both the same v 1
Remember that the magnitude of the relative velocity has to be equal before and after the collision!
In case 1 the bowling ball will almost remain at rest, and the golf ball will bounce back with speed close to v. v 2 2v
In case 2 the bowling ball will keep going with speed close to v, hence the golf ball will rebound with speed close to 2v.

34. Elastic Collisions II m v v M a) zero
Carefully place a small rubber ball (mass m) on top of a much bigger basketball (mass M) and drop these from the same height h so they arrive at the ground with the speed v. What is the velocity of the smaller ball after the basketball hits the ground, reverses direction, and then collides with the small rubber ball?
a) zero
b) v
c) 2v
d) 3v
e) 4v m v v M

35. Elastic Collisions II v 3v (a) (b) (c) m M a) zero
Carefully place a small rubber ball (mass m) on top of a much bigger basketball (mass M) and drop these from the same height h so they arrive at the ground with the speed v. What is the velocity of the smaller ball after the basketball hits the ground, reverses direction, and then collides with the small rubber ball?
a) zero
b) v
c) 2v
d) 3v
e) 4v
Remember that relative velocity has to be equal before and after collision! Before the collision, the basketball bounces up with v and the rubber ball is coming down with v, so their relative velocity is –2v. After the collision, it therefore has to be +2v!! v 3v
(a)
(b)
(c) m M

36. Elastic Collisions in 2-D
Two-dimensional collisions can only be solved if some of the final information is known, such as the final velocity of one object

37. CEBAF at JLab Precision hadronic microscopy
CEBAF at Jefferson Lab
recirculation through continuous-wave superconducting RF linacs
simultaneous beam delivery to 3 experimental halls with large complementary spectrometers
cold RF = stable, clean, quiet
up to 200 microAmps per hall, E ~ GeV, >80% polarization
Let me briefly introduce CEBAF
This photograph shows the characteristic “racetrack” shape of two cold RF linacs connected by recirculating arcs.
There are three experimental halls with complementary standard spectrometer equipment. Beam can be simultaneously delivered to all 3.
The machine operates in continuous wave mode, which maximizes the useful luminosity.
The cold RF is key in producing a stable, quiet and clean beam.
With up to 200 microAmps to a hall, an energy range of ~ GeV, and beam polarization routinely over 80%, there has never been an instrument like this for precision hadron microscopy.
This facility has over 1200 outside users, performing a wide range of studies.
An ideal machine for precision hadronic microscopy!
3 experimental Halls
First experiments begun in 1994
User community 1200 members strong

39. v1 v0 = 3.5x105 m/s 37o 53o initial v2 final
A proton collides elastically with another proton that is initially at rest. The incoming proton has an initial speed of 3.5x105 m/s and makes a glancing collision with the second proton. After the collision one proton moves at an angle of 37o to the original direction of motion, the other recoils at 53o to that same axis. Find the final speeds of the two protons.
v0 = 3.5x105 m/s
initial
37o
53o v2 v1
final

40. v1 v0 = 3.5x105 m/s 37o 53o initial v2 final
A proton collides elastically with another proton that is initially at rest. The incoming proton has an initial speed of 3.5x105 m/s and makes a glancing collision with the second proton. After the collision one proton moves at an angle of 37o to the original direction of motion, the other recoils at 53o to that same axis. Find the final speeds of the two protons.
Momentum conservation:
v0 = 3.5x105 m/s
initial
37o
53o v2 v1
final
if we’d been given only 1 angle, would have needed conservation of energy also!

41. Center of Mass Treat extended mass as a bunch of small masses:
In a uniform gravitational field you can treat gravitational force as if it acts at the “Center of Mass”

42. The center of mass is closer to the more massive object.
The center of mass of a system is the point where the system can be balanced in a uniform gravitational field.
For two objects:
The center of mass is closer to the more massive object.

43. Center of Mass
In general: Symmetry often very useful in determining the Center of Mass

44. The center of mass need not be within the object

45. Motion about the Center of Mass
The center of mass of a complex or composite object follows a trajectory as if it were a single particle - with mass equal to the complex object, and experiencing a force equal to the sum of all external forces on that complex object

46. Motion of the center of mass
Action/Reaction pairs inside the system cancel out

47. The total mass multiplied by the acceleration of the center of mass is equal to the net external force
The center of mass accelerates just as though it were a point particle of mass M acted on by

48. Momentum of a composite object

49. Recoil Speed
A cannon sits on a stationary railroad flatcar with a total mass of kg. When a 10-kg cannonball is fired to the left at a speed of 50 m/s, what is the speed of the center of mass (of the flatcar + cannonball)?
a) 0 m/s
b) 0.5 m/s to the right
c) 1 m/s to the right
d) 20 m/s to the right
e) 50 m/s to the right
Answer: b

50. Recoil Speed
A cannon sits on a stationary railroad flatcar with a total mass of kg. When a 10-kg cannonball is fired to the left at a speed of 50 m/s, what is the speed of the center of mass (of the flatcar + cannonball)?
a) 0 m/s
b) 0.5 m/s to the right
c) 1 m/s to the right
d) 20 m/s to the right
e) 50 m/s to the right
Because the initial momentum of the system was zero, the final total momentum must also be zero, regardless of the release of internal energy, internal forces, etc.
If no external forces act, the motion of the center of mass does not change

51. Recoil Speed II
A cannon sits on a stationary railroad flatcar with a total mass of kg. When a 10-kg cannonball is fired to the left at a speed of 50 m/s, what is the recoil speed of the flatcar?
a) 0 m/s
b) 0.5 m/s to the right
c) 1 m/s to the right
d) 20 m/s to the right
e) 50 m/s to the right
Answer: b

52. Recoil Speed II
A cannon sits on a stationary railroad flatcar with a total mass of kg. When a 10-kg cannonball is fired to the left at a speed of 50 m/s, what is the recoil speed of the flatcar?
a) 0 m/s
b) 0.5 m/s to the right
c) 1 m/s to the right
d) 20 m/s to the right
e) 50 m/s to the right
Because the initial momentum of the system was zero, the final total momentum must also be zero. Thus, the final momenta of the cannonball and the flatcar must be equal and opposite.
pcannonball = (10 kg)(50 m/s) = 500 kg-m/s
pflatcar = 500 kg-m/s = (1000 kg)(0.5 m/s)

53. Center of Mass a) higher b) lower c) at the same place
The disk shown below in (1) clearly has its center of mass at the center.
Suppose the disk is cut in half and the pieces arranged as shown in (2). Where is the center of mass of (2) as compared to (1) ?
a) higher
b) lower
c) at the same place
d) there is no definable CM in this case
Answer: a
(1)
X CM
(2)

54. Center of Mass CM a) higher b) lower c) at the same place
The disk shown below in (1) clearly has its center of mass at the center.
Suppose the disk is cut in half and the pieces arranged as shown in (2).
Where is the center of mass of (2) as compared to (1) ?
a) higher
b) lower
c) at the same place
d) there is no definable CM in this case
The CM of each half is closer to the top of the semicircle than the bottom. The CM of the whole system is located at the midpoint of the two semicircle CMs, which is higher than the yellow line.
(1)
X CM
(2) CM