Chapter 9 - Collisions
•
Momentum and force
•
Conservation of momentum
•
Impulse
•
Inelastic collisions
– Perfectly inelastic
•
Elastic collisions in one dimension
– moving target
– stationary target
•
Elastic collisions in two
dimensions
•
Center of mass
Momentum
• Linear momentum
– quantity of motion
– Product of mass times velocity
• The time rate of change of
the momentum of an object
is equal to the resulting net
external force acting on the
object.
p  mv
dp
 F  dt
Conservation of momentum
• If there are no external forces
• We say momentum is conserved
dp
 F  dt  0
p  constant
• For two particles we write:
p1i  p2i  p1f  p2f
m1v1i  m2 v2i  m1v1f  m2 v2f
Problem 1 Inelastic Collision
• Car 1 with a mass of 1000 kg and a velocity
of 20 m/s runs into the rear end of a larger
car with mass of 2000 kg initially at rest.
The two cars stick together.
• Find the final velocity
• Find the energy lost in the collision
Applications of conservation of
momentum
Impulse
dp
F
dt
dp  Fdt
p  I
tf
I   Fdt
ti
Average force during a collision
tf
I   Fdt  Ft
ti
Problem 2
• A ball (mass = 0.1 kg) is released from 2
meters and rebounds to 1.5 meters. What is
the Impulse of the floor on the ball
The ballistic pendulum
If you can measure M, m, and h,
how fast was the bullet traveling?
Elastic vs. inelastic
• Momentum is conserved
in all collisions.
• Elastic collision –
Kinetic energy is also
conserved.
• Inelastic collision –
Kinetic energy is not
conserved.
• Perfectly Inelastic –
Objects stick together
after the collision.
Elastic collisions
Momentum:
Energy:
m1v1i  m2 v2i  m1v1f  m2 v2f
1
1
1
1
2
2
2
2
m1v1i  m 2 v 2i  m1v1f  m 2 v 2f
2
2
2
2
Elastic collisions
m1  v1i  v1f   m2  v2f  v2i 
2
m1  v1i2  v1f2   m 2  v 2f
 v 2i2 
m1  v1i  v1f  v1i  v1f   m2  v2f  v2i  v2f  v2i 
 v1i  v1f    v2f  v2i 
v1i  v2i  v2f  v1f
Elastic collisions – equal mass
m1v1i  m2 v2i  m1v1f  m2 v2f
v1i  v2i  v1f  v2f
v1i  v2i  v2f  v1f
v1i  v2f
v2i  v1f
Elastic collision – mass at rest
v1
m1
m2
m1v1i  m1v1f  m2 v2f
m1  v1i  v1f   m2 v2f
v1i  v2i  v2f  v1f
v1i  v2f  v1f
2m1
v 2f  v1i
m1  m 2
m1  m 2
v1f  v1i
m1  m 2
Elastic collision – general case
v1
m1
v2
m2
2m1
m 2  m1
v 2f  v1i
 v 2i
m1  m 2
m1  m 2
m1  m 2
2m 2
v1f  v1i
 v 2i
m1  m 2
m1  m 2
Problem 3 Elastic Collision
• A 3 kg mass moving at 8 m/s in the x
direction collides with a 5 kg mass initially
at rest
• Find the final velocity of each mass.
• Find the final kinetic energy of each mass
m1
m2
Elastic collision in two dimensions
m2 is at rest
m1v1i  m1v1f  m2 v2f
m1v1i  m1v1f cos 1f  m2 v2f cos 2f
0  m1v1f sin 1f  m2 v2f sin 2f
1
1
1
2
2
m1v1i  m1v1f  m 2 v 22f
2
2
2
Problem 4
• Two shuffleboard disks of equal mass are involved in a
elastic glancing collision. One disk is initially at rest and is
struck by the other which is moving with a speed of 4 m/s.
After the collision, the incident disk moves along a
direction that makes an angle of 30o with its initial direction
of motion. The originally stationary disk moves in a
direction perpendicular to the final direction of motion of
the other disk. Find the final speeds.
30o
4 m/s
60o