1

+x direction m1 m2

v1i v2i

Before collision

During collision for

v1f v2f

After collision

Net momentum conservation for collision on frictionless horizontal surface

F on m2 from m1 F on m1 from m2

t

From Newton’s 3rd law, on m1 from m2 on m2 from m1F F

Multiplying on both sides, t for m1 for m2I I

2 2 for m2f ip p I Using I-M Theorem, 1 1 for m1f ip p I

1 2 1 for m1 2 for m2 1 2f f i i i ip p p I p I p p

2

System of objects

m1

m2

m3 The system: m1,

m2 and m3;

however m4 and

m5 are external

objects!

We consider system of “multiple” objects, which we can choose any way convenient for us.

m4

m5

For example,

3

m1

m2

m3

m4

m5

Internal vs. External Forces Internal forces: Forces between objects within the system

External forces: Forces from outside the system

4

m1

m2

m3 The system: m1,

m2 and m3;

however m4 and

m5 are external

objects!

m4

m5

1v2v

3v4v

5v

Net Momentum of System of Particles (or, Objects)

1 2 1 1 2 2... ...netP p p m v m v

5

m1

m2

m3

m4

m5

Net external force, Net external impulse

sum of all “external” forces acting on the system

, ,1 ,2 ...net ext ext extF F F Net external force :

system

, ,net ext net extI F t Net external impulse:

6

Impulse-Momentum Theorem for System of Particles

, , ,net ext net f net iI P P

Proof: use Newton’s 3rd law (see textbook)

Note: Internal forces do not matter for net momentum change.

Net external impulse is equal to net momentum change.

7

Conservation of momentum

, 0net extI

, ,net f net iP P

If , for example,

, 0net extF

If the net external impulse is zero, for example,

if the net external force is zero,

the net momentum of a system is conserved.

or t is small

8

Example

+x direction 3 kg particle 1 kg particle

1 m/s 4 m/s

Before collision

During collision

1 m/s ??? m/s

After collision

Two objects on a frictionless horizontal surface collide as shown above. We choose two objects as a system.

Find the velocity of the 1 kg object.

9

Example

+x direction 3 kg particle 1 kg particle

1 m/s 4 m/s

Before collision

During collision

??? m/s

After collision

Two objects on a frictionless horizontal surface collide and they move together after collision.

Find the velocity after collision.

glue

10

Example

+x direction 3 kg particle 1 kg particle

1 m/s ??? m/s

Before explosion

After explosion

Find the velocity of 1 kg object after explosion.

vi=0

11

Example 1: The archer

An archer stands at rest on frictionless ice and fires a 0.50 kg arrow horizontally at 50.0 m/s. The combined mass of the archer and bow is 60.0 kg. With what velocity does the archer move across the ice after firing the arrow?

12

Elastic vs. Inelastic Collisions

In a collision within a system of particles

Before collision

After collision

Net kinetic energy: 2 2

1 2 1 2

1 1... ...

2 2netK K K mv mv

If , ,net i net fK K elastic collision.

, ,net i net fK KIf inelastic collision.

, ,net i net fK KIf and objects move together after collision perfectly inelastic collision.

Elastic Collisions in 1-Dimension

Both total momentum & total kinetic energy are conserved.

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

1 1 2 2 1 1 2 2

1 1 1 1

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

+x direction

Before collision

After collision

v1i v2i

v1f=? v2f=?

m1 m2

&

and 1 2 2

1, 1, 2,

1 2 1 2

2f i i

m m mv v v

m m m m

1 2 12, 1, 2,

1 2 1 2

2f i i

m m mv v v

m m m m

(see text for proof)

iClicker Quiz In 1D elastic collision, if m1=m2, vi1=+10 m/s, v2i=0,

then, after collision v1f= _____ and v2f= _____.

a) 5 m/s; 5 m/s b) -5 m/s; 5 m/s c) 0 m/s; 10 m/s d) 0 m/s; -10 m/s e) -10 m/s; 0 m/s

and 1 2 2

1, 1, 2,

1 2 1 2

2f i i

m m mv v v

m m m m

1 2 12, 1, 2,

1 2 1 2

2f i i

m m mv v v

m m m m

+x direction

Before collision

After collision

V1i=10m/s V2i=0

v1f=? v2f=?

m1 m2

Collisions in Two Dimensions

If net external force is zero, net momentum is conserved.

, ,net i net fP P

x m1 m2

v1

v2

V1

V2

y Or

If net kinetic energy is conserved Elastic collision not conserved Inelastic collision

If moving together after collision Perfectly Inelastic collision (Kinetic energy is not conserved.)

, , , ,net i x net f xP P

, , , ,net i y net f yP P

1 1 2 2 ...netP m v m v , where

Example 2: Collision at an intersection

A 1500 kg car traveling east with a speed of 25 m/s collides at an intersection with a 2500 kg van traveling north at a speed of 20 m/s. Find the direction and magnitude of the velocity of the wreckage after the collision, assuming the vehicles stick together after the collision.

Example 1: 90 degree deflection rule in a game of pool

Assume that the collision is elastic, and the two balls have the same mass.

Show that the angle between the outgoing balls is 90 degree.

1,iv

2, fv

1, fv

m

m

(No forward, back or side spin is in effect.)