1. Physics 207, Lecture 12, Oct. 15 Assignment: HW5 due Wednesday
Agenda: Finish Chapter 9, start Chapter 10
Chapter 9: Momentum &
Collisions
Momentum conservation in 2D
Impulse
Assignment:
HW5 due Wednesday
HW6 posted soon 1

2. Impulse & Linear Momentum
Transition from forces to conservation laws
Newton’s Laws  Conservation Laws
Conservation Laws  Newton’s Laws
They are different faces of the same physics phenomenon for special “cases”

3. Lecture 12, Exercise 1 Momentum Conservation
Two balls of equal mass are thrown horizontally with the same initial velocity. They hit identical stationary boxes resting on a frictionless horizontal surface.
The ball hitting box 1 bounces elastically back, while the ball hitting box 2 sticks.
In which case does the box ends up moving fastest ?
No external force then, vectorially, COM
Box 1
Box 2
Same 1 2

4. Lecture 12, Exercise 1 Momentum Conservation
Which box ends up moving fastest ?
Examine the change in the momentum of the ball.
In the case of box 1 the balls momentum changes sign and so its net change is largest. Since momentum is conserved the box must have the largest velocity to compensate.
(A) Box (B) Box (C) same 2 1

5. A perfectly inelastic collision in 2-D
Consider a collision in 2-D (cars crashing at a slippery intersection...no friction). V v1
m1 + m2 m1 m2 v2
before
after
If no external force momentum is conserved.
Momentum is a vector so px, py and pz

6. Elastic Collisions Elastic means that the objects do not stick.
There are many more possible outcomes but, if no external force, then momentum will always be conserved
Start with a 1-D problem.
Before
After

7. Elastic Collision in 1-Dm1 m2
before
v1b
v2b x m2 m1
after
v1a
v2a

8. Force and Impulse (A variable force applied for a given time)
Gravity: usually a constant force to an object
Springs often provide a linear force (-k x) towards its equilibrium position (Chapter 10)
Collisions often involve a varying force
F(t): 0  maximum  0
We can plot force vs time for a typical collision. The impulse, J, of the force is a vector defined as the integral of the force during the time of the collision.

9. Force and Impulse (A variable force applied for a given time)
J reflects momentum transfer t ti tf t F
Impulse J = area under this curve !
(Transfer of momentum !)
Impulse has units of Newton-seconds

10. Force and Impulse
Two different collisions can have the same impulse since J depends only on the momentum transfer, NOT the nature of the collision. F t
same area F t t t
t big, F small
t small, F big

11. Average Force and Impulset
Fav F
Fav t t t
t big, Fav small
t small, Fav big

12. Example from last time
A 2 kg cart initially at rest on frictionless horizontal surface is acted on by a 10 N horizontal force along the positive x-axis for 2 seconds what is the final velocity?
F is in the x-direction F = ma so a = F/m = 5 m/s2
v = v0 + a t = 0 m/s + 2 x 5 m/s = 10 m/s (+x-direction)
but mv = F t [which is the area with respect to F(t) curve]

13. Lecture 12, Exercise 2 Force & Impulse
Two boxes, one heavier than the other, are initially at rest on a horizontal frictionless surface. The same constant force F acts on each one for exactly 1 second.
Which box has the most momentum after the force acts ? F
light
heavy
heavier
lighter
same
can’t tell

14. Lecture 12, Exercise 2 Force & Impulse
Two boxes, one heavier than the other, are initially at rest on a horizontal frictionless surface. The same constant force F acts on each one for exactly 1 second.
Which box has the most momentum after the force acts ?
(A) heavier (B) lighter (C) same F
light
heavy

15. Boxers:

16. Back of the envelope calculation
(1) marm~ 7 kg (2) varm~7 m/s (3) Impact time t ~ 0.01 s
 Impulse J = p ~ marm varm ~ 49 kg m/s
 F ~ J/t ~ 4900 N
(1) mhead ~ 6 kg
 ahead = F / mhead ~ 800 m/s2 ~ 80 g !
Enough to cause unconsciousness ~ 40% of fatal blow

17. Woodpeckers ahead ~ 600 - 1500 g How do they survive?
During "collision" with a tree 
ahead ~ g
How do they survive?
Jaw muscles act as shock absorbers
Straight head trajectory reduces damaging rotations (rotational motion is very problematic)

18. Energy What do we mean by an isolated system ?
What do we mean by a conservative force ?
If a force acting on an object act for a period of time then we have an Impulse  change (transfer) of momentum
What if we consider this force acting over a distance:
Can we identify another useful quantity?

19. y(t) = y0 + vy0 t + ½ ay t2 Dy = y(t)-y0= vy0 t + ½ ay t2
Energy
Fy = m ay and let the force be constant
y(t) = y0 + vy0 t + ½ ay t2 Dy = y(t)-y0= vy0 t + ½ ay t2
vy (t) = vy0 + ay t  t = (vy - vy0) / ay = Dvy / ay
So Dy = vy0 Dvy / ay ½ ay (Dvy/ay)2
= (vy vy0 - vy02 ) / ay + ½ (vy2 - 2vy vy0+vy02 ) / ay
2 ay Dy = (vy2 - vy02 )
Finally: may Dy = ½ m (vy2 - vy02 )
If falling: -mg Dy = ½ m (vy2 - vy02 )

20. -mg (yf – yi) = ½ m ( vyf2 -vyi2 )
Energy
-mg Dy= ½ m (vy2 - vy02 )
-mg (yf – yi) = ½ m ( vyf2 -vyi2 )
A relationship between y displacement and y speed
Rearranging
½ m vyi2 + mgyi = ½ m vyf2 + mgyf
We associate mgy with the “gravitational potential energy”

21. Energy ½ m vi2 + mgyi = ½ m vf2 + mgyf
Notice that if we only consider gravity as the external force then
then the x and z velocities remain constant
To ½ m vyi2 + mgyi = ½ m vyf2 + mgyf
Add ½ m vxi2 + ½ m vzi2 and ½ m vxf2 + ½ m vzf2
½ m vi2 + mgyi = ½ m vf2 + mgyf
where vi2 = vxi2 +vyi2 + vzi2
½ m v2 terms are referred to as kinetic energy

22. Emech is called “mechanical energy”
If only “conservative” forces are present, the total energy (sum of potential, U, and kinetic energies, K) of a system is conserved.
Emech = K + U
Emech = K + U = constant
K and U may change, but E = K + Umech remains a fixed value.
Emech is called “mechanical energy”

23. Another example of a conservative system: The simple pendulum.
Suppose we release a mass m from rest a distance h1 above its lowest possible point.
What is the maximum speed of the mass and where does this happen ?
To what height h2 does it rise on the other side ? v h1 h2 m

24. Example: The simple pendulum.
What is the maximum speed of the mass and where does this happen ?
E = K + U = constant and so K is maximum when U is a minimum. y
y=h1
y=0

25. Example: The simple pendulum.
What is the maximum speed of the mass and where does this happen ?
E = K + U = constant and so K is maximum when U is a minimum
E = mgh1 at top
E = mgh1 = ½ mv2 at bottom of the swing y
y=h1 h1
y=0 v

26. Example: The simple pendulum.
To what height h2 does it rise on the other side?
E = K + U = constant and so when U is maximum again (when K = 0) it will be at its highest point.
E = mgh1 = mgh2 or h1 = h2 y
y=h1=h2
y=0

27. Lecture 12, Exercise 3 Conservation of Mechanical Energy
A block is shot up a frictionless 40° slope with initial velocity v. It reaches a height h before sliding back down. The same block is shot with the same velocity up a frictionless 20° slope.
On this slope, the block reaches height 2h h
h/2
Greater than h, but we can’t predict an exact value.
Less than h, but we can’t predict an exact value.

28. Lecture 12, Example The Loop-the-Loop … again
To complete the loop the loop, how high do we have to let the release the car?
Condition for completing the loop the loop: Circular motion at the top of the loop (ac = v2 / R)
Use fact that E = U + K = constant !
Ub=mgh
Recall that “g” is the source of the centripetal acceleration and N just goes to zero is the limiting case.
Also recall the minimum speed at the top is
Car has mass m
U=mg2R
h ? R
(A) 2R (B) 3R (C) 5/2 R (D) 23/2 R

29. Lecture 12, Example The Loop-the-Loop … again
Use E = K + U = constant
mgh + 0 = mg 2R + ½ mv2
mgh = mg 2R + ½ mgR = 5/2 mgR
h = 5/2 R (C)
h ? R
(A) 2R (B) 3R (C) 5/2 R (D) 23/2 R

30. Inelastic collision in 1-D: Example 1
A block of mass M is initially at rest on a frictionless horizontal surface. A bullet of mass m is fired at the block with a muzzle velocity (speed) v. The bullet lodges in the block, and the block ends up with a speed V.
What is the initial energy of the system ?
What is the final energy of the system ?
Is energy conserved? x v V
before
after

31. Inelastic collision in 1-D: Example 1
What is the momentum of the bullet with speed v ?
What is the initial energy of the system ?
What is the final energy of the system ?
Is momentum conserved (yes)?
Is energy conserved? Examine Ebefore-Eafter v
No! V x
before
after

32. Example – Fully Elastic Collision
Suppose I have 2 identical bumper cars.
One is motionless and the other is approaching it with velocity v1. If they collide elastically, what is the final velocity of each car ?
Identical means m1 = m2 = m
Initially vGreen = v1 and vRed = 0
COM  mv = mv1f + mv2f  v1 = v1f + v2f
COE  ½ mv12 = ½ mv1f2 + ½ mv2f2  v12 = v1f2 + v2f2
v12 = (v1f + v2f)2 = v1f2 +2v1fv2f + v2f2  2 v1f v2f = 0
Soln 1: v1f = 0 and v2f = v Soln 2: v2f = 0 and v1f = v1

33. Lecture 12, Exercise for home Elastic Collisions
I have a line of 3 bumper cars all touching. A fourth car smashes into the others from behind. Is it possible to satisfy both conservation of energy and momentum if two cars are moving after the collision?
All masses are identical, elastic collision.
(A) Yes (B) No (C) Only in one special case v
Before v1 v2
After?

34. Finish Chapter 10, Start 11 (Work)
Physics 207, Lecture 12, Oct. 15
Agenda: Finish Chapter 9, start Chapter 10
Chapter 9: Momentum &
Collisions
Momentum conservation in 2D
Impulse
Assignment:
HW5 due Wednesday
HW6 posted soon
Finish Chapter 10, Start 11 (Work) 1

35. Lecture 12, Exercise for home Elastic Collisions
COM  mv = mv1 + mv2 so v = v1 + v2
COE  ½ mv2 = ½ mv12 + ½ mv22
v2 = (v1 + v2)2 = v12 + v22  v1 v2 = 0
(A) Yes (B) No (C) Only in one special case
Before
After?

36. Example of the U - F relationship
For a Hooke’s Law spring,
U(x) = (1/2)kx2
Notice that the derivative gives Hooke’s Law
Fx = - d ( (1/2)kx2) / dx
Fx = - 2 (1/2) kx
Fx = -kx

37. Equilibrium U U Example Spring: Fx = 0 => dU / dx = 0 for x=0
The spring is in equilibrium position
In general: dU / dx = 0  for ANY function establishes equilibrium U U
stable equilibrium
unstable equilibrium

38. Hooke’s Law Springs: Another kind of potential energy

39. Elastic vs. Inelastic Collisions
A collision is said to be elastic when energy as well as momentum is conserved before and after the collision Kbefore = Kafter
Carts colliding with a perfect spring, billiard balls, etc. vi
A collision is said to be inelastic when energy is not conserved before and after the collision, but momentum is conserved Kbefore  Kafter
Car crashes, collisions where objects stick together, etc.

40. 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 (friction)
Bending of metal and deformation
Kinetic energy is not conserved since negative work is by a non-conservative force done during the collision !
Momentum along a certain direction is conserved when there are no external forces acting in this direction.
In general, easier to satisfy than energy conservation.

41. Example of 2-D Elastic collisions: Billiards
If all we are given is the initial velocity of the cue ball, we don’t have enough information to solve for the exact paths after the collision. But we can learn some useful things...

42. Consider the case where one ball is initially at rest.
Billiards
Consider the case where one ball is initially at rest.
after
before
pa q pb
vcm
Pa f F
The final direction of the red ball will depend on where the balls hit.

43. x-dir COM: m vb = m va cos q + m Vb cos f
Billiards: All that really matters is conservation of energy and momentum
COE: ½ m vb2 = ½ m va2 + ½ m Va2
x-dir COM: m vb = m va cos q + m Vb cos f
y-dir COM: = m va sin q + m Vb sin f
pa q pb F
Pa f
before
after
vcm
Active Figure
The final directions are separated by 90° : q – f = 90°

44. Lecture 12 – Exercise 4 Pool Shark
Can I sink the red ball without scratching (sinking the cue ball) ?
(Ignore spin and friction)
(A) Yes (B) No (C) More info needed

45. Lecture 12 – Exercise 4 Pool Shark
Can I sink the red ball without scratching (sinking the cue ball) ?
(Ignore spin and friction)
(A) Yes (B) No (C) More info needed