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Momentum, Impulse and Collisions
Momentum
everyday connotations?
physical meaning
the “true” measure of motion (what changes in response to


applied forces)
dv d

F  m  (mv )
dt dt
Momentum (specifically Linear Momentum) defined


p  mv
to be generalized later
so
 dp
F 
dt
note: momentum is a vector
px = mvx , py = mvy , pz = mvz
Phys211C8 p1
Impulse
During a constant force


v
F  m
t




Ft  mv  mv  p

define Impulse J


J  Ft


J  p
for a force which is not constant :


t2

J   Fdt  p
t1
 

so J  Fav t  p

t2

t1 Fdt  t1 madt

t2
dv
  m dt
t1
dt



 mv 2  mv1  p
t2
Fx
(Fav)x
t
Phys211C8 p2
Integral is area under the curve!
For a particle initially at rest

p1  0


so p2  J
the particle’s momentum equals the impulse that accelerated the
particle from rest to its current state of motion.
(analogous to K = work to accelerate particle from rest…)
Kinetic energy can be written in terms of momentum and mass
1 2 m1 2
K  mv 
mv
2
m2
1 2 2
1

mv 
(mv) 2
2m
2m
p2
K
2m
Phys211C8 p3
Consider 2 particles of the same mass, the second with twice the speed of the first.
How do their momenta compare?
How do their Kinetic Energies compare?
Consider 2 particles with the same speed, the second with twice the mass of the first.
How do their momenta compare?
How do their Kinetic Energies compare?
Consider 2 particles initially at rest with the same size force acting for the same amount
of time, the second with twice the mass of the first.
How do their momenta compare?
How do their Kinetic Energies compare?
Consider 2 particles initially at rest with the same size force acting through the same
distance, the second with twice the mass of the first.
How do their momenta compare?
How do their Kinetic Energies compare?
Phys211C8 p4
Ex: A 0.40 kg ball impacts a wall horizontally with a speed of 30 m/s and rebounds
horizontally with a speed of 20 m/s. The ball is in contact with the wall for .01s.
Determine the impulse and average force on the ball.
Ex: A 0.40 kg soccer ball traveling horizontally to the left at 20 m/s is kicked up and to the
right at a 45 degree angle with a speed of 30 m/s. The ball is in contact with the foot of the
kicker for .01s.
Determine the impulse and average force on the ball.
Phys211C8 p5
Conservation of Momentum: an application of action-reaction
2 interacting objects, no external forces




dp A
dp B
FBonA 
, FAonB 
dt
dt


FBonA  FAonB
 

P  p A  pB



dP d 
dp A d p B

 p A  p B  

dt dt
dt
dt


 FBonA  FAonB  0

P is constant!
Generalize and consider external forces:
If the vector sum of external forces on a system is zero, the total
momentum of the system is constant.
Phys211C8 p6
mAvA1 + mBvB1 = mAvA2 + mBvB2
Using conservation of momentum in problems:
determine if momentum is conserved
select a coordinate system (momentum is a vector!)
sketch before and after diagrams
relate total initial momentum to total final momentum,
component by component!
solve equations (use additional equations as appropriate such
as conservation of energy)
Phys211C8 p7
Example: Rifle recoil: A 3 kg rifle is used to fire a 5 g bullet. The velocity of the bullet
relative to the ground is 300 m/s after being fired.
What is the momentum and energy of the bullet?
What is the momentum and energy of the rifle?
What is the recoil speed of the rifle?
Example: 1-D collision: 2 carts collide head-on on a frictionless track. The first cart has a
mass of .5 kg and approaches the collision with a speed of 2 m/s. The second cart has a mass
of .3 kg and approaches the collision with a speed of 2 m/s. After the collision, the second
cart rebounds from the collision at a speed of 2 m/s. Determine the initial kinetic energies
and final velocities and kinetic energies of both masses. How much energy is lost in this
collision?
Phys211C8 p8
Example: 2-D collision. A 5.00 kg mass initially moves in the positive x-direction with a
speed of 2.00 m/s, and then collides with a 3.00 kg mass which is initially at rest. After the
collision, the first mass is found to be moving at 1.00 m/s 30º from the positive x-axis.
What is the final velocity of the second mass?
What is the total initial and final kinetic energy of the system?
Phys211C8 p9
Elastic and Inelastic Collisions
Elastic Collisions
interaction is conservative force
mechanical energy is conserved
no “stickiness”
Inelastic Collisions
interaction is not conservative force
some mechanical energy is lost
some “stickiness”
Completely Inelastic Collisions
interaction is not conservative force
maximum loss of mechanical energy
colliders stick together after the collision
In all collisions, momentum is conserved; in elastic collisions,
energy is conserved as well.
Phys211C8 p10
Completely inelastic collisions
vA2 = vB2 = v2 so
mAvA1 + mBvB1 = (mA+ mB)v2
take object B initially at rest (can consider as 1-d problem)
m Av1  (m A  mB )v2
mA
so v2 
v1
m A  mB
1
2
K1  m Av1
2
2
 mA  2
1
1
2
K 2  (m A  mB )v2  (m A  mB )
 v1
2
2
 m A  mB 
so
K2
mA

K1 m A  m B
 energy is always lost in a completely inelastic collision
Phys211C8 p11
Example: 1-D collision: 2 carts collide head-on on a frictionless track. The first cart has a
mass of .5 kg and approaches the collision with a speed of 2 m/s. The second cart has a
mass of .3 kg and approaches the collision with a speed of 2 m/s. After the collision, the the
carts stick together. Determine the initial kinetic energies and final velocity and kinetic
energy of both masses. How much energy is lost in this collision?
Phys211C8 p12
Example: Ballistic Pendulum. A bullet of mass m is fired into a block of wood of mass M,
where it remains imbedded. The block is suspended like a pendulum, and swings up to a
maximum height y. Relate M, m and y to the bullets initial velocity.
Example: A 2000 kg car traveling east at 10 m/s collides (completely inelastically) with a
1000 kg car traveling north at 15 m/s. Find the velocity of the wreckage just after the
collision, and the energy lost in the collision.
Phys211C8 p13
Elastic Collisions
examine 1-d elastic collision, with B at rest before collision
m Av A1  0  m Av A 2  mB vB 2 
 momentum
 m A (v A1  v A 2 )  mB vB 2 
1
1
1
2
2
2
m Av A1  0  m Av A 2  mB vB 2 
2
2
2
 energy
2
2
2

 m A (v A1  v A 2 )  mB vB 2
 v A1  v A 2  vB 2
 v A2
or v A1  vB 2  v A 2
(relative velocity )
m A  mB
2m A

v A1 , vB 2 
v A1
m A  mB
m A  mB
Phys211C8 p14
Example: 1-D collision: 2 carts collide elastically head-on on a frictionless track. The first
cart has a mass of .5 kg and approaches the collision with a speed of 2 m/s. The second cart
has a mass of . 3 kg and approaches the collision with a speed of 2 m/s. Determine the initial
kinetic energies and final velocities and kinetic energies of both masses.
Example: A neutron (mass 1 u = 1.66E-27 kg) traveling at 2.6E7 m/s strikes a carbon
nucleus (mass 12 u). What are the velocities after the collision? by what factor is the
neutron’s kinetic energy reduced by the collision?
Phys211C8 p15
Example: A spacecraft of mass 825 kg approaches Saturn “head on” with an initial speed of
9.6 km/s while Saturn (mass 5.69E26 kg) moves along its orbit at 10.4 km/s. The
gravitational force of Saturn on the spacecraft swings the spacecraft back in the opposite
direction. What is the final speed of the spacecraft.
Example: An elastic (2-d) collision of two pucks on a frictionless air table occurs with the
first mass ( 0.500 kg) approaching at 4.00 m/s in the positive x-direction and the second
mass (0.300 kg) initially at rest. After the collision, the first puck moves off at a speed of
2.00 m/s in an unknown direction. What is the direction of the first pucks velocity after the
collision, and what is the speed and direction of the second puck after the collision/
Phys211C8 p16
Center of Mass
aka Center of Inertia
“average” location of mass on a system of particles
m1 x1  m2 x2   mi xi
xcm 

( similarly ycm , zcm )
m1  m2  
mi



m1r1  m2 r2   mi ri

rcm 

m1  m2  
mi
motion of center of mass




m
v
m v  m2 v 2  

i i
vcm  1 1

m1  m2  
mi




Mvcm  m1v1  m2 v 2    P
(total momentum)
Phys211C8 p17
External forces and the motion of center of mass






Macm  m1a1  m2 a 2    F  Fext  Fint

action  reaction Fint  0


dP

Macm  Fext 
dt


dv
d ( Mv cm )

since Macm  M cm 
dt
dt
example: A 50.0 kg woman walks from one end of 5m, 40.0 kg canoe to the other. Both the
canoe and the woman are initially at rest. If the friction between the water and the canoe is
negligible, how far does the woman move relative to shore? How far does the boat move
relative to shore?
Phys211C8 p18