THE IMMACULATE RECEPTION

MOMENTUMAP Physics C: Mechanics

WHAT IS MOMENTUM?

What is its definition?

How do we calculate it?

When do we use this term?

Why was this word invented?

What do we already know about it?

What do we want to know about it?

WHAT IS MOMENTUM?

What is its definition?

Momentum: the product of an object’s mass and its velocity

Momentum: “mass in motion”

Momentum: “quantity of motion”-

Newton

Momentum: It is a vector!

Momentum: is sometimes called linear momentum

WHAT IS MOMENTUM?

How do we calculate it?

p m

v

What are its units?

mass length time

kgm

s

px mvx

If object is moving in arbitrary direction:

py mvy

pz mvz

WHAT DO WE KNOW ABOUT MOMENTUM?

WHAT IS MOMENTUM? Why was this word invented? When do we use this term?

We are yet to make a distinction between a rhino moving at 5m/s and a hummingbird moving at 5m/s.

Thus far, how have we handled forces that are only briefly applied such as

collisions?(we pretended that doesn’t happen)

Some believed that this quantity is conserved in our universe.

HOW IS MOMENTUM RELATED TO OTHER PHYSICS CONCEPTS THAT WE HAVE ALREADY STUDIED?

F m

a

mdv

dt

dp

dt

We will soon see that it has many things in common with Energy, Newton’s 3rd law, and The

Calculus.

The time rate of change of linear momentum of a particle is equal to the net force acting on the

particle.

PAUSE TO THINK ABOUT CALCULUS CONCEPTS: Why is a derivative involved?

What does this say about the slope of a momentum-time graph?

The area under which graph might be meaningful?

So, how might an integral be involved?

Momentum may be changing non-uniformly with time

The slope of a momentum-time graph is net force!

The area under a force-time graph is a change in momentum!

The integral of force with respect to time is a change in momentum!

PAUSE TO THINK ABOUT CALCULUS CONCEPTS:

The integral of force with respect to time is a change in momentum!

F

dp

dt

F dt d

p

F dt d

p

F dt

p

I

F dt

t i

t f p

We call the left-hand side of this

equation the IMPULSE of the

force

PAUSE TO THINK ABOUT CALCULUS CONCEPTS:

The slope of a momentum-time graph is net force!

The area under a force-time graph

is a change in momentum or

an impulse

IMPULSE-MOMENTUM THEOREM:

I

F t

p

The impulse of a force F equals the change in momentum of the particle.

This is another way of saying that a net force must be applied to change an objects state of

motion.Why does this look different from the last

equation?

Because the force might be

constant!

A FEW THINGS ABOUT IMPULSE:

It is a vector in the same direction as the change in momentum.

It is not a property of an object! It is a measure of the degree to which a force changes a

particles momentum. We say an impulse is given to a particle.

What are its units?From the equation we see that they must be

the same as momentum’s units (kgm/s).

Impulse approximation: assume the force is applied only for an instant and that it is much

greater than other forces present.

ANOTHER QUESTION PLEASE…

TO STOP A SPEEDING TRAIN: EXPLAIN THESE VIDEOS IN PHYSICS TERMS.

QUICK CONCEPTUAL QUIZ

Can a hummingbird have more momentum than a rhino?

Why might an out of control truck hit a haystack or barrels and pile of sand as opposed to a wall as an emergency stop?

How is a ninja’s ability to break stacks of wood related to impulse and momentum?

What good is it to know an object’s momentum?

Question 2: If a boxer is able to make his impact time 5x longer by “riding” with the punch, how much will the impact force be reduced?

By 5x

Ft mv

Ft

t

mv

t

F mv

t

When a dish falls, will the impulse be less if it lands on a carpet than if it lands on a hard floor?

No – the same impulse – the force exerted on the dish is less because the time of momentum change increases.

EXAMPLES

Examples of Increasing Impact Time to decrease Impact Force:

Bend knees when jumping Gymnasts and wrestlers use mats

Glass dish falling on carpet rather than concrete

Acrobat safety net

Other examples???

OBSERVING CHANGES IN MOMENTUM:

CONSIDER TWO PARTICLES THAT CAN INTERACT, BUT ARE OTHERWISE ISOLATED FORM THEIR SURROUNDINGS.

What do we know about a collision between these two particles?

Newton’s law says that they exert equal and opposite forces on each other regardless of

comparative size (mass).

Is it possible for one particle to be in contact with the second particle for a longer period of

time than the second on the first?

No, so the impulse imparted on each must be the same.

THEREFORE…

THE PARTICLES MUST UNDERGO THE SAME CHANGES IN MOMENTUM!

Let’s look at this mathematically.

F2on1 dp1

dt

F1on 2 dp2

dt

1 2dp dp

dt dt

1 2 0dp dp

dt dt 1 2 0

dp p

dt

d

dtp1 p2 0

ptot p1 p2

dptot

dt0

What does it mean, conceptually, for a time

derivative of momentum to be zero

It means that the total momentum of the system is

constant over time.

aka Momentum is Conserved!

THE LAW OF CONSERVATION OF MOMENTUM

When two isolated, uncharged particles interact with each other, their total

momentum remains constant.

OR

The total momentum of an isolated system at all times equals its initial

momentum (before and after collisions).

p1i p2i p1f p2f

FIND THE REBOUND SPEED OF A 0.5 KG BALL FALLING STRAIGHT DOWN THAT HITS THE FLOOR MOVING AT 5M/S, IF

THE AVERAGE NORMAL FORCE EXERTED BY THE FLOOR ON THE BALL WAS 205N FOR 0.02S.

I Ft p

FN Fg t m v v0

v 205N 5N 0.02s

0.5kg 5m/s

v 3m/s

A) v/3 to the left B) The piece is at

rest. C) v/4 to the left D) 3v/4 to the left E) v/4 to the right

A mass m is moving east with speed v on a smooth horizontal surface explodes into two

pieces. After the explosion, one piece of mass 3m/4 continues in the same direction with

speed 4v/3. Find the magnitude and direction for the velocity of the other piece.

pbefore pafter

mv 3m

4

4v

3

1

m

4v

2

mv mv m

4v2

HOW GOOD ARE BUMPERS? A car of mass 1500kg is crash-tested into a

wall. It hits the wall with a velocity of -15m/s and bounces off with a velocity of 2.6m/s. If the collision lasts for 0.15s, what is the average force exerted on the car?

I m v v0 I 1500kg 2.6m/s ( 15m/s) I 2.64 104 kgm/s

I Ft

F 2.64 104 kgm/s

0.15s

F 1.76 105N

TYPES OF COLLISIONS

Energy is always conserved but may change types (mv2/2, mgh, kx2/2 etc). There is only one type of momentum (mv). We identify

collisions based upon their conservation of kinetic energy.

Inelastic•kinetic

energy is NOT constant

Elastic•kinetic

energy IS constant

INELASTIC COLLISIONS

These collisions are considered PERFECT when the objects collide and combine to

move as one object. Inelastic•Objects bounce but may be deformed so kinetic energy is transformed.

Perfectly Inelastic•Objects stick together

PERFECTLY INELASTIC COLLISIONS:

p1i p2i p12f

m1v1 m2v2 m1 m2 vf

ELASTIC COLLISIONS (IDEALLY)

m1v1i m2v2i m1v1f m2v2f

1

2m1v1i

2 1

2m2v2i

2 1

2m1v1f

2 1

2m2v2f

2

FOR ELASTIC COLLISIONS, FIND AN EXPRESSION FOR RELATIVE SPEED OF THE OBJECTS BEFORE AND AFTER COLLISION.

m1v1i m2v2i m1v1f m2v2f

From momentum conservation…

m1v1i m1v1f m2v2f m2v2i

m1 v1i v1f m2 v2f v2i

FOR ELASTIC COLLISIONS, FIND AN EXPRESSION FOR FINAL SPEED IN TERMS OF INITIAL SPEEDS AND MASS.

From kinetic energy conservation…

1

2m1v1i

2 1

2m2v2i

2 1

2m1v1f

2 1

2m2v2f

2

m1 v1i2 v1f

2 m2 v2f2 v2i

2 Divide out ½ and move like mass terms to the same side so mass can be factored out…

m1 v1i v1f v1i v1f m2 v2f v2i v2f v2i Factor difference of squares…

m1 v1i v1f v1i v1f m2 v2f v2i v2f v2i

m1 v1i v1f m2 v2f v2i

Combine our two results…

v1i v1f v2f v2i

v1i v2i v1f v2f

v1i v2i v1f v2f

The relative speed of the two objects before an elastic collision equals the negative of

their relative speed after.

SOLVE FOR FINAL SPEEDS IN TERMS OF INITIAL SPEEDS AND MASS.

1

2m1v1i

2 1

2m2v2i

2 1

2m1v1f

2 1

2m2v2f

2

m1v1i m2v2i m1v1f m2v2f

TWO-DIMENSIONAL COLLISIONS

Set coordinate system up with x-direction the same as one of the initial velocities

Label vectors in a sketch Write expressions for components of

momentum before and after collision for each object

v1i

v1f

v2f

v2fcosφ

v1fsinθ

-v2fsinφ

φ

v1fcosθθ

1 1 1 1 2 2cos cosi f fm v m v m v

1 1 2 20 sin sinf fm v m v

THE TYPES OF COLLISIONS ARE TREATED THE SAME MATHEMATICALLY.

pi pf