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PHYSICS 231
INTRODUCTORY PHYSICS I
Lecture 8
Last Lecture
•
Work for nonconstant force
•
Spring force
•
Potential Energy of Spring
•
Power
F  kx
W KE
P

t
t
P  Fv

Fx
x
1 2
PE  kx
2
Chapter 6
Momentum and Collisions
Momentum
r
p  mv
Definition:
Newton’s
2nd

Law:
v
Fm
t
p
 F
t
Conservation of Momentum
True for isolated particles (no external forces)
Proof:
Recall F12=-F21, (Newton’s 3rd Law)
F12  F21  0
p1 p2


0
t
t
p
i
 p1  p2  0
p1 f  p2 f  p1i  p2i
for isolated particles never changes!

Momentum is a Vector quantity
• Both Spx and Spy are conserved
px  mvx
py  mvy
Example 6.1
An astronaut of mass 80 kg
pushes away from a space
station by throwing a 0.75kg wrench which moves with
a velocity of 24 m/s relative
to the original frame of the
astronaut. What is the
astronaut’s recoil speed?
0.225 m/s
Center of mass does not accelerate
Xcm
Xcm
m1 x1  m2 x2  m3 x3  ...

(m1  m2  m3  ...)
m1x1  m2 x2  m3 x3  ...

(m1  m2  m3  ...)
m1 (x1 / t)  m2 (x2 / t)  m3 (x3 / t)  ...
 t 
(m1  m2  m3  ...)
p1  p2  p3  ...
 t 
(m1  m2  m3  ...)
 0 if total P is zero
Example 6.2
Ted and his ice-boat (combined mass = 240 kg) rest on the
frictionless surface of a frozen lake. A heavy rope (mass
of 80 kg and length of 100 m) is laid out in a line along
the top of the lake. Initially, Ted and the rope are at
rest. At time t=0, Ted turns on a wench which winds 0.5
m of rope onto the boat every second.
a) What is Ted’s velocity just after the wench turns on?
0.125 m/s
b) What is the velocity of the rope at the same time?
-0.375 m/s
c) What is the Ted’s speed just as the rope finishes?
0
d) How far did the center-of-mass of Ted+boat+rope move
0
e) How far did Ted move?
12.5 m
f) How far did the center-of-mass of the rope move?
-37.5 m
Example 6.3
A 1967 Corvette of mass
1450 kg moving with a
velocity of 100 mph
(= 44.7 m/s) slides on a
slick street and collides
with a Hummer of mass
3250 kg which is parked
on the side of the street.
The two vehicles interlock
and slide off together.
What is the speed of the
two vehicles immediately
after they join?
13.8 m/s =30.9 mph
Impulse
Impulse  Ft  p
Useful for sudden changes where the exact details of
the force are difficult to determine
For nonconstant F,
Impulse = Area under F vs. t curve
Bungee Jumper Demo
Example 6.4
A pitcher throws a 0.145-kg baseball
so that it crosses home plate
horizontally with a speed of 40 m/s.
It is hit straight back at the pitcher
with a final speed of 50 m/s.
a) What is the impulse delivered to
the ball?
b) Find the average force exerted by
the bat on the ball if the two are in
contact for 2.0 x 10–3 s.
c) What is the acceleration
experienced by the ball?
a) 13.05 kgm/s
b) 6,525 N c) 45,000 m/s2
Collisions
• Momentum is always conserved in a collision
• Classification of collisions:
• ELASTIC
• Both energy & momentum are conserved
• INELASTIC
• Momentum conserved, not energy
• Perfectly inelastic -> objects stick
• Lost energy goes to heat
Examples of Perfectly
Inelastic Collisions
• Catching a baseball
• Football tackle
• Cars colliding and sticking
• Bat eating an insect
Examples of Perfectly
Elastic Collisions
• Superball bouncing
• Electron scattering
Ball Bounce Demo
Example 6.5a
A superball bounces off the floor,
A)
B)
C)
D)
The net momentum of the earth+superball is conserved
The net energy of the earth+superball is conserved
Both the net energy and the net momentum are conserve
Neither are conserved
Example 6.5b
A astronaut floating in space catches a baseball
A)
B)
C)
D)
Momentum of the astronaut+baseball is conserved
Mechanical energy of the astronaut+baseball is conserve
Both mechanical energy and momentum are conserved
Neither are conserved
Example 6.5c
A proton scatters off another proton. No new
particles are created.
A)
B)
C)
D)
Net momentum of two protons is conserved
Net kinetic energy of two protons is conserved
Both kinetic energy and momentum are conserved
Neither are conserved
Perfectly Inelastic collision in 1-dimension
m1v1i  m2v2i  m1  m2 v f
•

Final velocities are the same
Example 6.6
A 5879-lb (2665 kg) Cadillac Escalade going 35 mph
=smashes into a 2342-lb (1061 kg) Honda Civic also
moving at 35 mph=15.64 m/s in the opposite
direction.The cars collide and stick.
a) What is the final velocity of the two vehicles?
b) What are the equivalent “brick-wall” speeds for
each vehicle?
a) 6.73 m/s = 15.1 mph
b) 19.9 mph for Cadillac, 50.1 mph for Civic
Example 6.7
A proton (mp=1.67x10-27 kg) elastically collides with a
target proton which then moves straight forward. If
the initial velocity of the projectile proton is 3.0x106
m/s, and the target proton bounces forward, what are
a) the final velocity of the projectile proton?
b) the final velocity of the target proton?
0.0
3.0x106 m/s
Elastic collision in 1-dimension
1. Conservation of Energy:
1
2
m1v1i2  12 m2v 2i2  12 m1v12f  12 m2v 22 f
(1)
2. Conservation of Momentum:
m1v1i  m2v 2i  m1v1 f  m2v 2 f
(2)

• Rearrange both equations and divide:

m1v1i2  v12f 
 m2 v 22 f  v 2i2 
m1v1i  v1 f v1i  v1 f  m2 v 2 f  v 2i v 2 f  v 2i 
m1v1i  v1 f 
 m2 v 2 f  v 2i 
v1i  v1 f  v 2 f  v 2i

(1)
 v1i  v 2i  v1 f  v 2 f 
(2)
Elastic collision in 1-dimension
Final equations for head-on elastic collision:
m1v1i  m2v 2i  m1v1 f  m2v 2 f
v1i  v 2i  v1 f  v 2 f 
•
Relative velocity changes sign
 • Equivalent to Conservation of Energy
Example 6.8
An proton (mp=1.67x10-27 kg) elastically collides with
a target deuteron (mD=2mp) which then moves straight
forward. If the initial velocity of the projectile
proton is 3.0x106 m/s, and the target deuteron
bounces forward, what are
a) the final velocity of the projectile proton?
b) the final velocity of the target deuteron?
vp =-1.0x106 m/s
vd = 2.0x106 m/s
Head-on collisions with heavier objects always lead to
reflections
Example 6.9a
The mass M1 enters from the left with velocity v0 and
strikes the mass M2=M1 which is initially at rest. The
collision is perfectly elastic.
a) Just after the collision v2 ______ v0.
A) >
B) <
C) =
Example 6.9b
The mass M1 enters from the left with velocity v0 and
strikes the mass M2=M1 which is initially at rest. The
collision is perfectly elastic.
Just after the collision v1 ______ 0.
A) >
B) <
C) =
Example 6.9c
The mass M1 enters from the left with velocity v0 and
strikes the mass M2=M1 which is initially at rest. The
collision is perfectly elastic.
Just after the collision P2 ______ M1v0.
A) >
B) <
C) =
Example 6.9d
The mass M1 enters from the left with velocity v0 and
strikes the mass M2=M1 which is initially at rest. The
collision is perfectly elastic.
At maximum compression, the energy stored
in the spring is ________ (1/2)M1v02
A) >
B) <
C) =
Example 6.9e
The mass M1 enters from the left with velocity v0 and
strikes the mass M2<M1 which is initially at rest. The
collision is perfectly elastic.
Just after the collision v2 ______ v0.
A) >
B) <
C) =
Example 6.9f
The mass M1 enters from the left with velocity v0 and
strikes the mass M2<M1 which is initially at rest. The
collision is perfectly elastic.
Just after the collision v1 ______ 0.
A) >
B) <
C) =
Example 6.9g
The mass M1 enters from the left with velocity v0 and
strikes the mass M2<M1 which is initially at rest. The
collision is perfectly elastic.
Just after the collision P2 ______ M1v0.
A) >
B) <
C) =
Example 6.9h
The mass M1 enters from the left with velocity v0 and
strikes the mass M2<M1 which is initially at rest. The
collision is perfectly elastic.
At maximum compression, the energy stored
in the spring is ________ (1/2)M1v02
A) >
B) <
C) =
Example 6.9i
The mass M1 enters from the left with velocity v0 and
strikes the mass M2>M1 which is initially at rest. The
collision is perfectly elastic.
Just after the collision v2 ______ v0.
A) >
B) <
C) =
Example 6.9j
The mass M1 enters from the left with velocity v0 and
strikes the mass M2>M1 which is initially at rest. The
collision is perfectly elastic.
Just after the collision v1 ______ 0.
A) >
B) <
C) =
Example 6.9k
The mass M1 enters from the left with velocity v0 and
strikes the mass M2>M1 which is initially at rest. The
collision is perfectly elastic.
Just after the collision P2 ______ M1v0.
A) >
B) <
C) =
Example 6.9l
The mass M1 enters from the left with velocity v0 and
strikes the mass M2>M1 which is initially at rest. The
collision is perfectly elastic.
At maximum compression, the energy stored
in the spring is ________ (1/2)M1v02
A) >
B) <
C) =