1. Lecture 12 Chapter 9: Momentum & Impulse
Goals:
Chapter 9: Momentum & Impulse
Solve problems with 1D and 2D Collisions
Solve problems having an impulse (Force vs. time)
Chapter 10
Understand the relationship between motion and energy
Define Potential & Kinetic Energy
Develop and exploit conservation of energy principle
Assignment:
HW6 due Wednesday 3/3
For Tuesday: Read all of chapter 10 1

2. Momentum Conservation
Momentum conservation (recasts Newton’s 2nd Law when net external F = 0) is an important principle (usually when forces act over a short time)
It is a vector expression so must consider Px, Py and Pz
if Fx (external) = 0 then Px is constant
if Fy (external) = 0 then Py is constant
if Fz (external) = 0 then Pz is constant

3. Inelastic collision in 1-D: Example
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 final speed V.
In terms of m, M, and V :
What is the momentum of the bullet with speed v ? x v V
before
after

4. Inelastic collision in 1-D: Example
What is the momentum of the bullet with speed v ?
Key question: Is x-momentum conserved ?
P Before
P After v V
before
after x

5. Exercise Momentum is a Vector (!) quantity
A block slides down a frictionless ramp and then falls and lands in a cart which then rolls horizontally without friction
In regards to the block landing in the cart is momentum conserved?
Yes No
Yes & No
Too little information given

6. Exercise Momentum is a Vector (!) quantity
x-direction: No net force so Px is conserved.
y-direction: Net force, interaction with the ground so
depending on the system (i.e., do you include the Earth?)
Py is not conserved (system is block and cart only)
2 kg
5.0 m
30°
Let a 2 kg block start at rest on a 30° incline and slide vertically a distance 5.0 m and fall a distance 7.5 m into the 10 kg cart
What is the final velocity of the cart?
10 kg
7.5 m

7. Exercise Momentum is a Vector (!) quantity
1) ai = g sin 30°
= 5 m/s2
2) d = 5 m / sin 30°
= ½ ai Dt2
10 m = 2.5 m/s2 Dt2
2s = Dt
v = ai Dt = 10 m/s
vx= v cos 30°
= 8.7 m/s
x-direction: No net force so Px is conserved
y-direction: vy of the cart + block will be zero and we can ignore vy of the block when it lands in the cart. i j N
5.0 m mg
30°
30°
Initial Final
Px: MVx + mvx = (M+m) V’x
M 0 + mvx = (M+m) V’x
V’x = m vx / (M + m)
= 2 (8.7)/ 12 m/s
V’x = 1.4 m/s
7.5 m y x

8. Home Exercise Inelastic Collision in 1-D with numbers
Do not try this at home!
ice
(no friction)
Before: A 4000 kg bus, twice the mass of the car, moving at 30 m/s impacts the car at rest.
What is the final speed after impact if they move together?

9. Home exercise Inelastic Collision in 1-D
v = 0
ice
M = 2m V0
(no friction)
initially
vf =?
finally
2Vo/3 = 20 m/s

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

11. A perfectly inelastic collision in 2-D
If no external force momentum is conserved.
Momentum is a vector so px, py and pz are conseved V v1
m1 + m2 q m1 m2 v2
before
after
x-dir px : m1 v1 = (m1 + m2 ) V cos q
y-dir py : m2 v2 = (m1 + m2 ) V sin q

12. 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

13. 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.

14. Billiards: Without external forces, conservation of momentum (and energy Ch. 10 & 11)
x-dir Px : m vbefore = m vafter cos q + m Vafter cos f
y-dir Py : = m vafter sin q + m Vafter sin f
before
after
pafter q pb
Pafter f F

15. Force and Impulse (A variable force applied for a given time)
Gravity: At small displacements a “constant” force t
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.

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

17. 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

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

19. 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

20. Boxing: Use Momentum and Impulse to estimate g “force”

21. Back of the envelope calculation
(1) marm~ 7 kg (2) varm~7 m/s (3) Impact time t ~ 0.01 s
 Question: Are these reasonable?
 Impulse J = p ~ marm varm ~ 49 kg m/s
 Favg ~ J/t ~ 4900 N
(1) mhead ~ 6 kg
 ahead = F / mhead ~ 800 m/s2 ~ 80 g !
Enough to cause unconsciousness ~ 40% of a fatal blow
Only a rough estimate!

22. Chapter 10: Energy (Forces over distance)
We need to define an “isolated system” ?
We need to define “conservative force” ?
Recall, chapter 9, force acting for a period of time gives an
impulse or a change (transfer) of momentum
What if a force acts over a distance:
Can we identify another useful quantity?

23. For Tuesday: Read all of chapter 10
Lecture 12
Assignment:
HW6 due Wednesday 3/3
For Tuesday: Read all of chapter 10