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

dpF

dt

Conservation of momentum

• If there are no external forces

• We say momentum is conserved

• For two particles we write:

dpF 0

dt

p constant

1i 2i 1f 2fp p p p

1 1i 2 2i 1 1f 2 2fm v m v m v m v

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

dpF

dt

dp Fdt

p I

f

i

t

t

I Fdt

Average force during a collision

f

i

t

t

I Fdt F t

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

1 1i 2 2i 1 1f 2 2fm v m v m v m v

2 2 2 21 1i 2 2i 1 1f 2 2f

1 1 1 1m v m v m v m v

2 2 2 2

Momentum:

Energy:

Elastic collisions

1 1i 1f 2 2f 2im v v m v v

2 2 2 21 1i 1f 2 2f 2im v v m v v

1 1i 1f 1i 1f 2 2f 2i 2f 2im v v v v m v v v v

1i 1f 2f 2iv v v v

1i 2i 2f 1fv v v v

Elastic collisions – equal mass

1 1i 2 2i 1 1f 2 2fm v m v m v m v

1i 2i 1f 2fv v v v

1i 2i 2f 1fv v v v

1i 2fv v 2i 1fv v

Elastic collision – mass at rest

m1 m2

1 1i 1 1f 2 2fm v m v m v 1 1i 1f 2 2fm v v m v

1i 2i 2f 1fv v v v 1i 2f 1fv v v

12f 1i

1 2

2mv v

m m

1 21f 1i

1 2

m mv v

m m

v1

Elastic collision – general case

m1 m2

v1 v2

1 2 12f 1i 2i

1 2 1 2

2m m mv v v

m m m m

1 2 21f 1i 2i

1 2 1 2

m m 2mv v v

m m m m

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

1 1i 1 1f 2 2fm v m v m v

1 1i 1 1f 1f 2 2f 2fm v m v cos m v cos

1 1f 1f 2 2f 2f0 m v sin m v sin

2 2 21 1i 1 1f 2 2f

1 1 1m v m v m v

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.

4 m/s

30o

60o