General Physics I Momentum

Linear Momentum:

• Definition: For a single particle, the momentum p is defined as:

p = mv

(p is a vector since v is a vector).

So px = mvx etc.

Units of linear momentum are kg m/s.

Momentum Conservation

• The concept of momentum conservation is one of the most fundamental principles in physics.

• This is a component (vector) equation.

– Momentum along a certain direction is conserved when there are no external forces acting in this direction.

• Recall that force causes a chance in velocity and as momentum is defined by velocity….well, you see!

• You will see that we often have momentum conservation even when energy is not conserved.

1-D Conservation of Linear Momentum

• What this means is that the momentum “before” and the momentum “after” must be equal to each other. – We will primarily deal with conservation of momentum with

respect to 1-D linear collision situations – For a 2-body system this is:

m1v1i + m2v2i = m1v1f + m2v2f

This is a component vector equation since velocity is a vector!

So don’t forget that this means that the velocity could be

positive or negative

Comment on Energy Conservation

• We have seen that the energy of a system is not necessarily conserved. – Energy can be lost:

• Heat (bomb) • Bending of metal (crashing cars)

• In terms of collisions, kinetic energy is not conserved since work is done during the collision! – We talk about kinetic energy since, in the context of

momentum, the object(s) has velocity.

1/2 m1v2

1i + 1/2 m2v2

2i = 1/2 m1v2

1f + 1/2 m2v2

2f

We will also deal with conservation of kinetic energy with respect to 1-D linear collision situations

For a 2-body system this is:

Elastic vs. Inelastic Collisions • A collision is said to be elastic when kinetic energy as

well as momentum is conserved before and after the collision. Kbefore = Kafter – Carts colliding with a spring in between, billiard balls, etc.

vi

A collision is said to be inelastic when kinetic energy is not conserved before and after the collision, but momentum is conserved. Kbefore Kafter

Car crashes, collisions where objects stick together, etc.

Inelastic Collision in 1-D • A block of mass M is initially at rest on a frictionless horizontal surface. A

bullet of mass m is fired at the block with a muzzle velocity (speed) v. The bullet lodges in the block, and the block ends up with a speed V. In terms of m, M, and V : – What is the initial energy of the system? – What is the final energy of the system?

– What does momentum conservation tell us?

– Is kinetic energy conserved?

v

V

before after

x

Momentum Conservation

• Two balls of equal mass are thrown horizontally with the same initial velocity. They hit identical stationary boxes resting on a frictionless horizontal surface.

• The ball hitting box 1 bounces back with the same velocity it had initially, while the ball hitting box 2 gets stuck. – Which box ends up moving faster?

(a) Box 1 (b) Box 2 (c) Same

1 2 V1 V2

Collisions

• A box sliding on a frictionless surface collides and sticks to a second identical box which is initially at rest.

• What is the ratio of initial to final kinetic energy of the system?

(a) 1

(b)

(c) 2

2

Inelastic collision in 2-D • Consider a collision in 2-D (cars crashing at a

slippery intersection...no friction).

v1

v2

V

before after

m1

m2

m1 + m2

Inelastic collision in 2-D... • There are no net external forces acting.

– Use momentum conservation for both components.

v1

v2

V = (Vx,Vy)

m1

m2

m1 + m2

fxox PP m v m m Vx1 1 1 2

Vm

m mvx

1

1 21

fyoy PP m v m m Vy2 2 1 2

Vm

m mvy

2

1 22

X:

y:

Explosion (inelastic un-collision)

Before the explosion: M

m1 m2

v1 v2

After the explosion:

No external forces, so P is conserved.

Initially: P =

Finally: P =

The Crazy Bomb

• A bomb explodes into 3 identical pieces. Which of the following configurations of velocities is possible?

(a) 1 (b) 2 (c) Both

m m

v V

v

m

m m

v v

v

m

(1) (2)

Elastic Collisions

• Elastic means that kinetic energy is conserved as well as momentum.

• This gives us more constraints – We can solve more complicated problems!! – Billiards (2-D collision) – The colliding objects

have separate motions after the collision as well as before.

• Let’s start with a simpler 1-D problem.....

Initial Final

Elastic Collision in 1-D

v1i v2i

initial

x

m1 m2

v1f v2f

final

m1 m2

Elastic Collision in 1-D

v1i v2i

v1f v2f

before

after

x

m1 m2

Conserve PX:

m1v1i + m2v2i = m1v1f + m2v2f

Conserve Kinetic Energy:

1/2 m1v2

1i + 1/2 m2v2

2i = 1/2 m1v2

1f + 1/2 m2v2

2f

Suppose we know v1i and v2i

We need to solve for v1f and v2f

Should be no problem 2 equations & 2 unknowns!

This relationship becomes: v1i – v2i = v2f – v1f

An example of 2-D elastic collisions: Billiards.

• If all we know is the initial velocity of the cue ball, we don’t have enough information to solve for the exact paths after the collision. But we can learn some useful things...

Billiards.

• Consider the case where one ball is initially at rest.

pf

pi

F Pf

initial final The final direction of the

red ball will depend on

where the balls hit.

vnet

Billiards.

• The final directions are separated by 90o.

pf

pi

F Pf

initial final

vnet

Billiards.

• So, we can sink the red ball without sinking the white ball.

Billiards. • So, we can sink the red ball without sinking the

white ball. • However, we can also scratch. All we know is

that the angle between the balls is 90o.

End of Momentum Lecture