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: HW5 due Wednesday For Wednesday: Read all of chapter 10 Physics 207: Lecture 12, Pg 1
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 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 2
Inelastic collision in 1-D: Example
What is the momentum of the bullet with speed
v
?
m
v Key question: Is x-momentum conserved ?
Before
m
v M 0
After
aaaa
(
m
M
) V
v
before
V
after Physics 207: Lecture 12, Pg 3
x
Exercise
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.
Which box ends up moving fastest ?
A.
Box 1 B.
Box 2 C.
same 1 2 Physics 207: Lecture 12, Pg 4
Exercise
Momentum Conservation
The ball hitting box 1 bounces elastically back, while the ball hitting box 2 sticks.
Which box ends up moving fastest ?
Notice the implications from the graphical solution: Box 1’s momentum must be bigger because the length of the summed momentum must be the same.
The longer the green vector the greater the speed Before
After
Before After Ball 1
Ball 1
Ball 2 Ball 2 Box 1
Box 1
Box 2 Box 2 Box 1+Ball 1 Box 1+Ball
1
Box 2+Ball 2 Box 2+Ball 2 1 2 Physics 207: Lecture 12, Pg 5
A perfectly inelastic collision in 2-D
Consider a collision in 2-D (cars crashing at a slippery intersection...no friction).
v
1
V
q
m 1 + m 2 m 1 m 2
v
2
before after If no external force momentum is conserved.
Momentum is a vector so p x , p y and p z Physics 207: Lecture 12, Pg 6
A perfectly inelastic collision in 2-D
If no external force momentum is conserved.
Momentum is a vector so p x , p y and p z are conseved
V v
1
q
m 1 + m 2 m 1 m 2
v
2
before x-dir p x : m 1 v 1 y-dir p y : m 2 v 2 =
(m
1 =
(m
1
+ m
2
+ m
2 after
) V cos
q
) V sin
q Physics 207: Lecture 12, Pg 7
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 8
Billiards
Consider the case where one ball is initially at rest. after before
p
b
p
a
q
F v
cm
P
a
f The final direction of the red ball will depend on where the balls hit.
Physics 207: Lecture 12, Pg 9
Billiards: All that really matters is conservation momentum (and energy Ch. 10 & 11)
Conservation of Momentum x-dir P y-dir P x y : : m v before = m v after 0 = m v after cos sin q q + m V after + m V after cos f sin f before
p
b
after
p
after
q
F P
after
f Physics 207: Lecture 12, Pg 10
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
its equilibrium position (Chapter 10) x) towards 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 11
Force and Impulse (A variable force applied for a given time)
J
reflects momentum transfer
J
t F
dt
t
(
d p
/
dt
)
dt
p
d p F
Impulse
J
= area under this curve !
(Transfer of momentum !) Impulse has units of Newton-seconds
t i
t t f
Physics 207: Lecture 12, Pg 12
t
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 F
same area
t
t
big,
F
small
t t
t
t
small,
F
big Physics 207: Lecture 12, Pg 13
Average Force and Impulse
F av
F
t
t
big,
F av
small
t
F av
F t
t
t
small,
F av
big Physics 207: Lecture 12, Pg 14
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 F heavy
A.
heavier B.
lighter C.
D.
same can’t tell Physics 207: Lecture 12, Pg 15
Boxing: Use Momentum and Impulse to estimate g “force”
Physics 207: Lecture 12, Pg 16
Back of the envelope calculation
J
t F
dt
F
avg
t
(1) m arm
~ 7 kg (2)
v arm
~7 m/s (3)
Impact time
t
~ 0.01 s Question: Are these reasonable?
Impulse J =
p
~ m arm v arm ~ 49 kg m/s
F ~ J/
t ~ 4900 N
(1) m head ~ 6 kg
a head = F / m head ~ 800 m/s 2 ~ 80 g !
Enough to cause unconsciousness ~ 40% of fatal blow Only a rough estimate!
Physics 207: Lecture 12, Pg 17
Woodpeckers
During "collision" with a tree
a
head
~ 600 - 1500 g How do they survive?
• Jaw muscles act as shock absorbers • Straight head trajectory reduces damaging rotations (rotational motion is very problematic) Physics 207: Lecture 12, Pg 18
Home Exercise
The only force acting on a 2.0 kg axis. Notice that the plot is
force
object moving along the x vs
time
. If the velocity v x is +2.0 m/s at 0 sec, what is v x at 4.0 s ? p = m v = Impulse m v = J 0,1 + J 1,2 + J 2,4 m v = (-8)1 N s + ½ (-8)1 N s + ½ 16(2) N s m v = 4 N s v = 2 m/s v x = 2 + 2 m/s = 4 m/s Physics 207: Lecture 12, Pg 19
Chapter 10: Energy
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 acting over a distance:
Can we identify another useful quantity?
Physics 207: Lecture 12, Pg 20
Energy
F y = m a y y(t) = y 0 and let the force be constant + v y0 t + ½ a y t 2 y = y(t)-y 0 = v y0 t + ½ a y t 2 v y (t) = v y0 + a y t Eliminate t and regroup So y = v y0 (v y - v y0 ) / a y t = (v + ½ a y y - v (v y 2 y0 ) / a - 2v y y v y0 +v y0 2 ) / a y 2 y = ( v y0 v y v y0 2 ) / a y + ½ y = ( v y0 2 ) / a y + ½ (v (v y y 2 2 2v y v y0 + v y0 2 + v y0 2 ) / a y ) / a y y = ½ (v y 2 v y0 2 ) / a y Physics 207: Lecture 12, Pg 21
Energy
And now
y =
½
(v
y 2
-
v y0 2
) / a
y can be rewritten as:
ma
y
y = ½ m (v
y 2
- v
y0 2
)
And if the object is falling under the influence of gravity then
a
y
= -g
Physics 207: Lecture 12, Pg 22
Energy
-
mg y= ½ m (v y 2 - v y0 2 )
-
mg ( y f – y i ) = ½ m ( v yf 2 -v yi 2 ) A relationship between
y-displacement
and change in the y-speed Rearranging to give initial on the left and final on the right ½ m v yi 2
+
mgy i = ½ m v yf 2
+
mgy f We now define mgy as the “gravitational potential energy ” Physics 207: Lecture 12, Pg 23
Energy
Notice that if we only consider gravity as the external force then the x and z velocities remain constant To ½ m v yi 2
+
mgy i = ½ m v yf 2
+
mgy f Add ½ m v xi 2 + ½ m v zi 2 and ½ m v xf 2 + ½ m v zf 2 ½ m v i 2
+
mgy i = ½ m v f 2
+
mgy f where v i 2 = v xi 2 +v yi 2 + v zi 2 ½ m v 2 terms are defined to be kinetic energies (A scalar quantity of motion) Physics 207: Lecture 12, Pg 24
Energy
If only “conservative” forces are present, the total energy ( sum of potential, U, and kinetic energies, K ) of a system is conserved
E mech = K + U E mech = K + U
=
constant
K
and
U
may change, but
E mech = K + U
remains a fixed value.
E
mech
is called “mechanical energy” Physics 207: Lecture 12, Pg 25
Example of a conservative system: The simple pendulum.
Suppose we release a mass
m
its lowest possible point.
from rest a distance
h 1
above What is the maximum speed of the mass and where does this happen ?
To what height
h 2
does it rise on the other side ?
h 1 m v h 2
Physics 207: Lecture 12, Pg 26
Example: The simple pendulum.
What is the maximum speed of the mass and where does this happen ?
E = K + U = constant is a minimum.
and so K is maximum when U
y y= h 1 y= 0
Physics 207: Lecture 12, Pg 27
Example: The simple pendulum.
What is the maximum speed of the mass and where does this happen ?
E = K + U = constant is a minimum and so K is maximum when U E = mgh 1 at top E = mgh 1 = ½ mv 2 at bottom of the swing
y y= h 1 y=0 h 1 v
Physics 207: Lecture 12, Pg 28
Example: The simple pendulum.
To what height
h2
does it rise on the other side?
E = K + U = constant again (when K = 0) and so when U is maximum it will be at its highest point.
E = mgh 1 = mgh 2 or h 1 = h 2
y y= h 1 =h 2 y=0
Physics 207: Lecture 12, Pg 29
1 st
Cart Exercise Revisited: How does this help?
Part: Find v at bottom of incline a i = g sin 30 ° = 5 m/s 2 j N E mech
is conserved
K i + U i 0 + mgy i (2gy) ½ = K f = ½ mv 2 + U + 0 = v = (100) ½ f m/s v x = v cos 30 ° = 8.7 m/s
One step, no FBG needed
i 5.0 m y mg 30 ° 7.5 m d = 5 m / sin 30 ° = ½ a i t 2 10 m = 2.5 m/s 2 t 2 2s = t v = a i t = 10 m/s v x = v cos 30 ° = 8.7 m/s x Physics 207: Lecture 12, Pg 30
Home exercise
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 A.
2h B.
h C.
D.
E.
h/2 Greater than
h
, but we can’t predict an exact value.
Less than
h
, but we can’t predict an exact value.
Physics 207: Lecture 12, Pg 31
Exercise 3: U, K, E & Path
1.
2.
3.
A ball of mass m, initially at rest, is released and follows three difference paths. All surfaces are frictionless Ball is dropped Ball slides down a straight incline Ball slides down a curved incline After traveling a vertical distance h , how do the speeds compare?
1 2 3 h (A) 1 > 2 > 3 (B) 3 > 2 > 1 (C) 3 = 2 = 1 (D) Can’t tell Physics 207: Lecture 12, Pg 32