The ___________________ of a particle of mass m and velocity v is defined as

The linear momentum is a vector quantity.

It’s direction is along v.

The components of the momentum of a particle:

Chapter 9: Linear Momentum and Collisions

Name___________________

Momentum Facts

• _________________

• Momentum is a _________ quantity!

• _________ and _________vectors point in the _____ direction.

• SI unit for momentum: ____________

• Momentum is a _________ quantity

• __________________________________________________

• Momentum is ________ proportional to both ____ and _____.

• Something big and slow could have the same momentum as something small and fast.

Momentum Examples10 kg

3 m /s10 kg

Note: The momentum vector does not have to be drawn 10 times longer than the velocity vector, since only vectors of the same quantity can be compared in this way.

5 g

9 km /s

p =26º

Equivalent Momenta

Bus: m = 9000 kg; v = 16 m /s p = 1.44 ·105 kg · m /s

Train: m = 3.6 ·104 kg; v = 4 m /s p = 1.44 ·105 kg · m /s

Car: m = 1800 kg; v = 80 m /s p = 1.44 ·105 kg · m /s

The train, bus, and car all have different masses and speeds, but their ___________ are the _______ in _________.

The difficulty in bringing each vehicle to rest--in terms of a combination of the force and time required--would be the _______, since they each have the same momentum.

Impulse DefinedImpulse is defined as the product force acting on an object and the time during which the force acts. The symbol for impulse is J. So, by definition:

Example: A 50 N force is applied to a 100 kg boulder for 3 s.

Note that we didn’t need to know the mass of the object in the above example.

Impulse UnitsJ = F t shows why the SI unit for impulse is the _________________

Impulse and momentum have the ___________________

______________________________

The impulse due to all forces acting on an object (the net force) is equal to the change in momentum of the object:

We know the units on both sides of the equation are the same (last slide), but let’s prove the theorem formally:

Imagine a car hitting a wall and coming to rest. The force on the car due to the wall is large (big F ), but that force only acts for a small amount of time (little t ). Now imagine the same car moving at the same speed but this time hitting a giant haystack and coming to rest. The force on the car is much smaller now (little F ), but it acts for a much longer time (big t ). In each case the impulse involved is the same since the change in momentum of the car is the same. Any net force, no matter how small, can bring an object to rest if it has enough time. A pole vaulter can fall from a great height without getting hurt because the mat applies a smaller force over a longer period of time than the ground alone would.

Stopping Time

F t = F t

Impulse - Momentum Example

A 1.3 kg ball is coming straight at a 75 kg soccer player at 13 m/s who kicks it in the exact opposite direction at 22 m/s with an average force of 1200 N. How long are his foot and the ball in contact?

During this contact time the ball compresses substantially and then decompresses. This happens too quickly for us to see, though. This compression occurs in many cases, such as hitting a baseball or golf ball.

Fnet vs. t graphFnet (N)

t (s)6

A variable strength net force acts on an object in the positive direction for 6 s, thereafter in the opposite direction. Since impulse is Fnet t, the area under the curve is equal to the impulse, which is the change in momentum. The net change in momentum is the area above the curve minus the area below the curve. This is just like a v

vs. t graph, in which net displacement is given area under the curve.

____________

As long as there are __________________ acting on a system of particles, collisions between the particles will exhibit conservation of linear momentum.

This means that the vector sum of the momenta ________ collision is _______ to the vector sum of the momenta of the particles _________.

Conservation of Momentum in 1-DWhenever two objects collide (or when they exert forces on each other without colliding, such as gravity) momentum of the system (both objects together) is conserved. This mean the total momentum of the objects is the same before and after the collision.

before: ______________

after: _______________

m1 m2

v1v2

m1 m2

va vb

Directions after a collisionOn the last slide the boxes were drawn going in the opposite direction after colliding. This isn’t always the case. If we solved the conservation of momentum equation (red box) for vb and got a negative answer, it would mean that m2 was still moving to the left after the collision. As long as we interpret our answers correctly, it matters not how the velocity vectors are drawn.

m1 v1 - m2 v2 = - m1 va + m2 vb

m1 m2

v1v2

m1 m2

va vb

Collision between two objects _______________________________

Collision between two objects__________________________________

Collision between two objects________________________________

Simple Examples of Head-On Collisions(___________________________________________)

Simple Examples of Head-On Collisions

Collision between two objects _______________________________

Collision between two objects__________________________________

Collision between two objects________________________________

(___________________________________________)

___________ is conserved in any collision, ________________________________________

_____________ is _____ conserved in elastic collisions.

_____________________________: After colliding, particles stick together. There is a loss of energy (deformation).

_____________________________: Particles bounce off each other without loss of energy.

_____________________________: Particles collide with some loss of energy, but don’t stick together.

Notice that p and v are vectors and, thus have a direction (+/-)

lossfii

flossi

Evmmvmvm

KEK

221

222

211 )(

2

1

2

1

2

1There is a loss in energy Eloss

For collisions in one dimension:

Suppose we know the initial masses and velocities.

Then:

Note, that these are pretty specialized equations, (elastic collision in one dimension, known initial velocities, and masses)

Black board example 9.2

Two carts collide elastically on a frictionless track. The first cart (m1 = 1kg) has a velocity in the positive x-direction of 2 m/s; the other cart (m = 0.5 kg) has velocity in the negative x-direction of 5 m/s.

(a) Find the speed of both carts after the collision.

(b) What is the speed if the collision is perfectly inelastic?

(c) How much energy is lost in the inelastic collision?

Black board example 9.3 and demoDetermining the speed of a bullet

A bullet (m = 0.01kg) is fired into a block (0.1 kg) sitting at the edge of a table. The block (with the embedded bullet) flies off the table (h = 1.2 m) and lands on the floor 2 m away from the edge of the table.

a.) What was the speed of the bullet?b.) What was the energy loss in the bullet-block collision? (skip)

vb = ?

h = 1.2 m

x = 2 m

Motion of a System of Particles.

Newton’s second law for a System of Particles

The center of mass of a system of particles (combined mass M) moves like one equivalent particle of mass M would move under

the influence of an external force.

zCMznetyCMynetxCMxnet

CMnet

MaFMaFMaF

aMF

,,,,,,

Center of mass

Center of mass for many particles:

Where is the center of mass of this arrangement of particles.

(m3 = 2 kg; m1 = m2 = 1 kg)?

Velocity of the center of mass: Acceleration of the center of mass:

Black board example 9.6

A rocket is shot up in the air and explodes.

Describe the motion of the center of mass before and after the explosion.

A method for finding the center of mass of any object.

- Hang object from two or more points.

- Draw extension of suspension line.

- Center of mass is at intercept of these lines.

A change in momentum is called “impulse”:

During a collision, a force F acts on an object, thus causing a change in momentum of the object:

f

i

t

t

dttFJp )(

For a constant (average) force:

Think of hitting a soccer ball: A force F acting over a time t causes a change p in the momentum (velocity) of the ball.

A soccer player hits a ball (mass m = 440 g) coming at him with a velocity of 20 m/s. After it was hit, the ball travels in the opposite direction with a velocity of 30 m/s.

(a) What impulse acts on the ball while it is in contact with the foot?

(b) The impact time is 0.1s. What average force is the acting on the ball?

(c) How much work was done by the foot? (Assume and elastic collision.) (skip)

Black board example 9.6

Sample Problem 1

7 kg

v = 0

700 m/s

A rifle fires a bullet into a giant slab of butter on a frictionless surface. The bullet penetrates the butter, but while passing through it, the bullet pushes the butter to the left, and the butter pushes the bullet just as hard to the right, slowing the bullet down. If the butter skids off at 4 cm/s after the bullet passes through it, what is the final speed of the bullet?(The mass of the rifle matters not.)

35 g

7 kgv = ?

35 g

4 cm/s

Sample Problem 27 kg

v = 0

700 m/s

35 g

Same as the last problem except this time it’s a block of wood rather than butter, and the bullet does not pass all the way through it. How fast do they move together after impact?

v7. 035 kg

Note: Once again we’re assuming a frictionless surface, otherwise there would be a frictional force on the wood in addition to that of the bullet, and the “system” would have to include the table as well.

Black board example 9.1

You (100kg) and your skinny friend (50.0 kg) stand face-to-face on a frictionless, frozen pond. You push off each other. You move backwards with a speed of 5.00 m/s.

(a) What is the total momentum of the you-and-your-friend system?

(b) What is your momentum after you pushed off?

(c) What is your friends speed after you pushed off?

(d) How much energy (work) did you and your friend expend?(skip)

Sample Problem 3

EarthM

apple

An apple is originally at rest and then dropped. After falling a short time, it’s moving pretty fast, say at a speed V. Obviously, momentum is not conserved for the apple, since it didn’t have any at first. How can this be?

m

F

F

V

v

m V = M v

Sample Problem 4

before

after

3 kg 15 kg10 m/s 6 m/s

3 kg 15 kg4.5 m/s v

A crate of raspberry donut filling collides with a tub of lime Kool Aid on a frictionless surface. Which way on how fast does the Kool Aid rebound?

After the collision the lime Kool Aid is moving _______________

Conservation of Momentum in 2-D

m1m2

v1

v2

m1 m2

vavb

12

a b

To handle a collision in 2-D, we conserve momentum in each dimension separately. Choosing down & right as positive: before:

after:

Conservation of momentum equations:

Conserving Momentum w/ Vectors

m1m2

m1 m2

12

a b

BEFORE

AFTER

This diagram shows momentum vectors, which are parallel to their respective velocity vectors. Note p1 + p 2

= p a + p b and p before = p after as conservation of momentum demands.

p before

p after

Exploding Bomb

A c m e

before

after

A

c

m

eA bomb, which was originally at rest, explodes and shrapnel flies every which way, each piece with a different mass and speed. The momentum vectors are shown in the after picture.

continued on next slide

Exploding Bomb (cont.)

Since the momentum of the bomb was zero before the explosion, it must be zero after it as well. Each piece does have momentum, but the total momentum of the exploded bomb must be zero afterwards. This means that it must be possible to place the momentum vectors tip to tail and form a closed polygon, which means the vector sum is zero.

If the original momentum of the bomb were not zero, these vectors would add up to the original momentum vector.

Two-dimensional collisions (Two particles)

Conservation of momentum:

Split into components:

If the collision is elastic, we can also use conservation of energy.

Velocity Components in Projectile Motion(In the absence of air resistance.)

Note that the ____________ component of the velocity remains the __________ if airresistance can be ignored.

Collision between two objects of the same mass. One mass is at rest.

Example of Non-Head-On Collisions

(Energy and Momentum are Both Conserved)

If you vector add the total momentum after collision,you get the total momentum before collision.

2-D Sample Problem

0.3 kg

34 m/s40

152 gA mean, old dart strikes an innocent mango that was just passing by minding its own business. Which way and how fast do they move off together?

before

after

5 m/s

452 g

v

Working in grams and taking left & down as + :

Dividing equations :

Substituting into either of the first two equations :

Black board example 9.5

Accident investigation. Two automobiles of equal mass approach an intersection. One vehicle is traveling towards the east with 29 mi/h (13.0 m/s) and the other is traveling north with unknown speed. The vehicles collide in the intersection and stick together, leaving skid marks at an angle of 55º north of east. The second driver claims he was driving below the speed limit of 35 mi/h (15.6 m/s).

13.0 m/s

??? m/s

a) Is he telling the truth?

b) What is the speed of the “combined vehicles” right after the collision?

c) How long are the skid marks (k = 0.5)?

• For every type of linear quantity we have a rotational quantity that does much the same thing

Linear Quantities

Rotational Quantities

ROTATIONAL INERTIA & ANGULAR MOMENTUM

Rotational Inertia(I) • AKA (not really but could be) Rotational

Mass

• _________________________________– Objects that are rotating about an axis tend to

stay rotating, objects not rotating tend to remain at rest, unless an outside torque is applied

• _________________________________

• It’s the rotational equivalent to mass,– Harder to give an ang. acc. to an object w/ a

larger I

Moment of InertiaAny moving body has inertia.

______________________________________________________________________________________________________________

Single point-like massm

rSystem of masses

r2

m2

r1

m1Q

Q

Moment of Inertia ExampleTwo merry-go-rounds have the same mass and are spinning with the same angular velocity. One is solid wood (a disc), and the other is a metal ring. Which has a bigger moment of inertia relative to its center of mass?

mm

r r

answer:

The big idea• Rotational Inertia depends _______________

• If either one of these is large, then rotational inertia is ______, and object will be harder to ______

• Different types of objects have different equations for rotational inertia

• But all equations have ___________ in them.

Rotational Inertia • Some objects have

more rotational inertia than others

__________________________________________________________________________________________________________________________________________________

Angular Momentum, L

r

v

m

Depends on ___________________________________ from a particular point.

If _____________________ then the magnitude of angular momentum w/ resp. to point Q is given by _____________________

In this case L points out of the page. If the mass were moving in the opposite direction, L would point into the page.

Unit: ___________

A _______ is needed to change L, just a _____ is needed to change p.

Anything spinning has angular has angular momentum. ____________________________________________________________________________________

Q

Angular Momentum: General DefinitionIf r and v are _________________ then the angle between these two vectors must be taken into account. The general definition of angular momentum is given by a vector cross product:

This formula works regardless of the angle. As you know from our study of cross products, the magnitude of the angular momentum of m relative to point Q is:

In this case, ______________________, L points out of the page. If the mass were moving in the opposite direction, L would point into the page.

r

v

m

Q

Comparison: Linear & Angular Momentum

Linear Momentum, p

• _________________________ ___________________________

•__________________________

• _________________________

•_____________________________________________________

•_____________________________________________________

•________________________________________________________________________________

Angular Momentum, L

• _________________________ ___________________________

•__________________________

• _________________________

•_____________________________________________________

•_____________________________________________________

•________________________________________________________________________________

Angular AccelerationAngular acceleration occurs when a spinning object spins faster or slower.

Note how this is very similar to a = v / t for linear acceleration.

Ex: If a wind turbine spinning at 21 rpm speeds up to 30 rpm over 10 s due to a gust of wind, its average angular acceleration is 9 rpm / 10 s. This means every second it’s spinning 9 revolutions per minute faster than the second before. Let’s convert the units:

Torque & Angular Acceleration

Newton’s 2nd Law, as you know, is _________

The 2nd Law has a rotational analog: ___________

A force is required for a body to undergo acceleration. A “turning force” (a torque) is required for a body to

undergo angular acceleration.

Both m and I are measures of a body’s inertia (resistance to change in motion).

Linear Momentum & Angular MomentumRecall, angular momentum’s magnitude is given by:

r

So, if a net torque is applied, angular velocity must change, which changes angular momentum.

proof:

So net torque is the rate of change of angular momentum, just as net force is the rate of change of linear momentum.

From the formula v = r , we get

v

m

Why does a tightrope walker carry a long pole?

• __________________________________________________________________________________

• _________________________________________

• _________________________________________

• http://www.youtube.com/watch?v=w8Tfa5fHr3s

Sports Connection

• Running–

• Gymnastics/Diving–

Angular Momentum• “inertia of rotation”

• Ang. Momentum= Rotational Inertia X Rotational Speed– L=Iω

Conservation of Angular Momentum• __________________________________

________________________________

• __________________________________________________________________

• __________________________________________________________________

I – represents rotational inertia

ω -represents angular speed

Sports Connection…• Ice skating

– – http://www.youtube.com/watch?v=AQLtcEAG9v0

– http://www.youtube.com/watch?v=NtEnEeEyw_s

–

–

• Gymnastics (pummel horse or floor routine)–

Do cats violate physical law?

• Video• _______________

__________________________________________________________________________________________________________________________________________________________

59

• Helicopter tail rotor failure

• Tail rotor failure #2

Universe Connection• Rotating star shrinks

radius…. What happens to rotational speed??–

• Rotating star explodes outward…. What happens to rotational speed??–

Make a list of equations