Chapter 6
Momentum and Collisions
Momentum
Definition:


p  mv
Important because it is CONSERVED
proof:



v p
F m

t t


Ft  p
Since F12=-F21,

 pi


p1  p2  0
for isolated particles never changes!
Vector quantity
px  mvx
p y  mv y
 Both Spx and Spy are conserved
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?
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
If total momentum is zero,


m1v1 m2v2  0
m1
r

m

r

0
1
2
2


m1r1  m2 r2  Constant
m1r1  m2 r2
 Constant
m1  m2
Do the back-to-back demo
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) What was Ted’s acceleration during this time?
-6.25x10-4 m/s2
e) How far did Ted move?
12.5 m
f) How far did the center-of-mass of the rope move?
-37.5 m
g) How far did the center-of-mass of Ted+boat+rope
0
move?
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 when not interested in
force but only effects of force
Bunjee Jumper Demo
Graphical Representation of Impulse
Impulse  Ft  p
For a complicated
force, impulse is
represented
graphically by the
area under the
F vs.t curve.
F
Total Impulse
F
t
t
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
Example 6.5
A typical 80-kg Physics
231 student (right)
drives into a concrete
wall at 30 mph = 13.4
m/s. The car buckles by
0.5 m as the car stops.
The air bag increases
the time of the collision;
It will also absorb some
of the energy from the
a) What
amount of time does it take to bounce back?
body It will spread out
(assume constant a)
the area of contact.
b) What is the average force exerted on the student?
(both in lbs and N)
a) 0.0746 s
b) 14,400 N = 31,700 lbs
Elastic & Inelastic 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
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
Ballistic Pendulum: used to measure speed
of bullet. 0.5-kg block of wood swings up
by height h = 65 cm after stopping 8.0-g
bullet.
What was bullet’s velocity?
227 m/s
Example 6.8
A 5-g bullet traveling at 500 m/s embeds in a 1.495
kg block of wood resting on the edge of a 0.9-m high
table. How far does the block land from the edge of
the table?
71.4 cm
Example 6.9
Tarzan (M=80 kg) swings on a 12m vine by letting go from an angle
of 60 degrees from the vertical.
At the bottom of his swing, he
picks up Jane (m=50 kg). To what
angle do Tarzan and Jane swing?
35.8 degrees
(indepedent of L or g)
Examples of Elastic Collisions
• pool balls
• electron scattering (sometimes)
• Earth-superball scattering
Ball Bounce Demo
Example 6.10
An 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
Equal-Mass Collision Demo
Example 6.11
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 proton 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
(Perfectly) Inelastic Collisions
in Two Dimensions
Two Equations : Two unknowns (vfx, vfy)
m1v1ix  m2v2ix  (m1  m2 )v fx
m1v1iy  m2v2iy  (m1  m2 )v fy
Example 6.12
A 1200-kg vehicle moving at 25.0
m/s east collides with a vehicle of
mass 1500 kg moving northward
at 20.0 m/s. After they join,
what is their final speed and
direction?
vf = 15.7 m/s
qf = 45
Elastic Collisions in Two Dimensions
Three Equations : Four Unknowns (v1f,v2f,q1f,q2f)
1 2 1 2 1 2 1 2
mv1i  mv2i  mv1 f  mv2 f
2
2
2
2
m1v1ix  m2v2ix  m1v1 fx  m2v2 fx
m1v1iy  m2v2iy  m1v1 fy  m2v2 fy
=0
Example 6.13
A projectile proton moving with v0=120,000 m/s
collides elastically with a second proton at rest.
If one proton leaves at an angle q=30,
a) what is its speed?
b) what are the speed and direction of the
second proton?
a) v1 = 103,923 m/s
b) v2 = 60,000 m/s q2 = 60
Working out answer in center-of-mass
1. Find c.o.m. velocity and subtract it from
both v1 and v2


m1v1,i  m2v2,i

vcom 
m

m
1


 2
Note:
v1,i  v1,i  vcom



v2,i  v2,i  vcom


m1v1,i  m2v2,i  0
2. Problem is easy to solve in this frame:
Perfectly elastic : v1 f  v1i , v2 f  v1 f
Perfectly inelastic : v1 f  v2 f  0
Working out answer in center-of-mass
3. Add back c.o.m. velocity to both v1 and v2



v1,i  v1,i  vcom



v2,i  v2,i  vcom
Example 6.14
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. ( >, < or =)
=
a) Just after the collision v2 ______
v0.
=
b) Just after the collision v1 ______
0.
c) At maximum compression, the energy stored
=
in the spring is ________
(1/2)M1v02
Example 6.15
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. ( >, < or =)
a)
b)
c)
d)
>
Just after the collision v2 ______
v0.
>
Just after the collision v1 ______
0.
<
Just after the collision p2 ______
M1v0.
At maximum compression, the energy stored
<
in the spring is ________
(1/2)M1v02
Example 6.16
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. ( >, < or =)
a)
b)
c)
d)
<
Just after the collision v2 ______
v0.
<
Just after the collision v1 ______
0.
>
Just after the collision p2 ______
M1v0.
At maximum compression, the energy stored
<
in the spring is ________
(1/2)M1v02