Transcript Physics 207: Lecture 2 Notes

Lecture 12
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
Physics 207: Lecture 12, Pg 1
Momentum Conservation

FEXT



dv d (mv) dP

 ma  m 

dt
dt
dt

and if FEXT  0


dP
 0 implies that P  constant
dt

P
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
Physics 207: Lecture 12, Pg 2
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
Physics 207: Lecture 12, Pg 3
Inelastic collision in 1-D: Example
What is the momentum of the bullet with speed v ?

mv
 Key question: Is x-momentum conserved ?
P After
P Before
mv  M 0  (m  M )V
aaaa
v
V
after
x
before
v  (1  M / m)V
Physics 207: Lecture 12, Pg 4
Exercise
Momentum is a Vector (!) quantity


A.
B.
C.
D.
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
Physics 207: Lecture 12, Pg 5
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
7.5 m into the 10 kg cart
10 kg
What is the final velocity of the cart?
Physics 207: Lecture 12, Pg 6
Exercise
Momentum is a Vector (!) quantity


1) ai = g sin 30°
= 5 m/s2
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
2) d = 5 m / sin 30°
lands in the cart.
j
= ½ ai Dt2
N
i 5.0 m
30° mg
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
10 m = 2.5 m/s2 Dt2
2s = Dt
v = ai Dt = 10 m/s
vx= v cos 30°
= 8.7 m/s
7.5 m
y
x
Physics 207: Lecture 12, Pg 7
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
Physics 207: Lecture 12, Pg 10
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
q
m1 + m2
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

Physics 207: Lecture 12, Pg 11
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
Physics 207: Lecture 12, Pg 12
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.
Physics 207: Lecture 12, Pg 13
Billiards: Without external forces, conservation
of momentum (and energy Ch. 10 & 11)



Conservation of Momentum
x-dir Px : m vbefore = m vafter cos q + m Vafter cos f
y-dir Py :
0
= m vafter sin q + m Vafter sin f
after
before
pafter q
pb
F
Pafter f
Physics 207: Lecture 12, Pg 14
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.

Physics 207: Lecture 12, Pg 15
Force and Impulse
(A variable force applied for a given time)

J a vector that reflects momentum transfer
 t

t
p 
J   F dt   (dp / dt )dt   dp
F
Impulse J = area under this curve !
(Transfer of momentum !)
t
Dt
Impulse has units of Newton-seconds
ti
tf
Physics 207: Lecture 12, Pg 16
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
same area
F
Dt
Dt big, F small
t
Dt
t
Dt small, F big
Physics 207: Lecture 12, Pg 17
Average Force and Impulse
F
Fav
F
Fav
Dt
Dt big, Fav small
t
Dt
t
Dt small, Fav big
Physics 207: Lecture 12, Pg 18
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
A.
B.
C.
D.
light
F
heavy
heavier
lighter
same
can’t tell
Physics 207: Lecture 12, Pg 19
Boxing: Use Momentum and Impulse to estimate g “force”
Physics 207: Lecture 12, Pg 20
Back of the envelope calculation
 t

J   F dt  Favg Dt
(1) marm~ 7 kg
(2) varm~7 m/s (3) Impact time Dt ~ 0.01 s
Question: Are these reasonable?
 Impulse J = Dp ~ marm varm ~ 49 kg m/s
 Favg ~ J/Dt ~ 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!
Physics 207: Lecture 12, Pg 21
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?
Physics 207: Lecture 12, Pg 22
Lecture 12

Assignment:
 HW6 due Wednesday 3/3
 For Tuesday: Read all of chapter 10
Physics 207: Lecture 12, Pg 23