Linear Momentum Lecturer: Professor Stephen T. Thornton

Linear Momentum

Lecturer:

Professor Stephen T. Thornton

A system of particles is known to have a total kinetic energy of zero. What can you say about the total momentum of the system?

A) momentum of the system is positiveB) momentum of the system is negativeC) momentum of the system is zeroD) you cannot say anything about the momentum of the system

Reading Quiz

A system of particles is known to have a total kinetic energy of zero. What can you say about the total momentum of the system?

A) momentum of the system is positiveB) momentum of the system is

negativeC) momentum of the system is zeroD) you cannot say anything about the

momentum of the system

Since the total kinetic energy is zero, this means that all of the particles are at rest (v = 0). Therefore, since nothing is moving, the total momentum of the system must also be zero.

Reading Quiz

Last Time

Conservation of Energy

Escape velocity

Power

Potential energy diagrams

TodayDefine linear momentum

Relationship between K.E. and momentum

More general form of 2nd law

Impulse

Internal and external forces

Collisions

New Concept – Linear Momentum

Linear momentum is simply the

product of mass and velocity.

Linear momentum is a vector.

Sometimes we say just “momentum”.

SI unit: kg · m/s

p mv

22

2 2

2

1 1

2 2

1

2 2

mvK mv m

m

p pK m

m m

Kinetic energy and linear momentum are intimately related. Remember this result:

2

2

pK

m

Do demo with bouncing ball and bean bag.

One recoils, the other doesn’t.

Change in Momentum

total 1 2 3

ˆ) 0

ˆ

ˆ ˆ)

ˆ2

Momentum is a vector

...

f i

i f

i f

p p p

a p mv p

p mv

b p mv p mv

p mv

p p p p

j

j

j j

j

Momentum and Newton’s 2nd Law

A more general form of Newton’s 2nd law:

netii

F F ma

net

dpF

dt

net

( ) , if mass is constant.

so if mass is constant.

f i f f i i

f i

p p p m v m v

m v v m v

p vm ma

t tp dp

F mat dt

Let’s see if the equations are consistent.

But when mass is not constant, our new general form should be used.

net

dpF

dt

ImpulseWhat is impulse and why is it useful?

Forces sometimes act between objects over very short times.

Examples:

Bouncing balls

Bat hitting a ball

Collisions

The Average Force During a Collision

Definition of ImpulseForce can vary considerably over the time of interaction, so let’s consider the average force, :avF

av

av

av

av

unit is N s

and f

i

t

t

J F t

pF

t

F t p

J F t Fdt p J

J

Impulse is just the change in momentum!

av

f

i

t

tJ F t Fdt p

Do egg in a sheet demo.

Conservation of Linear Momentum

What happens when = 0?

Then

netF

0 and momentum is conserved!

We have i f

p

p p

net

dpF

dt

If the net force acting on an object is zero, its momentum is conserved.

Do demos:Rocket bicycleReaction carsFire extinguisher rocket cartWater rocket2-liter bottle rocket

Conservation of Linear Momentum

Law of conservation of linear momentum:

When the net external force on a system of objects is zero, the total momentum of the system remains constant.

Equivalently,

The total momentum of an isolated system remains constant.

Conceptual QuizA) speeds upA) speeds upB) maintains B) maintains

constant speedconstant speedC) slows downC) slows downD) stops immediatelyD) stops immediately

An open cart rolls along a frictionless track while it is raining. As it rolls, what happens to the speed of the cart as the rain collects in it? (Assume that the rain falls vertically into the box.)

Because the rain falls in vertically, it adds no momentum to the box, thus the box’s momentum is conserved. However, because the mass of the box slowly increasesincreases with the added rain, its velocity has to decreasedecrease.

Conceptual QuizA) speeds upA) speeds upB) maintains B) maintains

constant speedconstant speedC) slows downC) slows downD) stops immediatelyD) stops immediately

An open cart rolls along a frictionless track while it is raining. As it rolls, what happens to the speed of the cart as the rain collects in it? (Assume that the rain falls vertically into the box.)

Follow-up:Follow-up: What happens to the cart when it stops raining? What happens to the cart when it stops raining?

Internal and External Forces

If we have a system of particles, then there can be internal forces (for example, that hold the object together).

Internal forces always occur in action-reaction pairs and the sum will be zero.

int

net ext

0F

F F

System of ObjectsInternal forces have no effect on the net momentum of an object.

If the net external force acting on a system is zero, then the net momentum is conserved.

Momentum of every particle in system is not conserved, only the net.

net 1 2 3 ... constantp p p p

Conservation of Momentum

Momentum conservation works for a rocket as long as we consider the rocket and its fuel to be one system and account for the mass loss of the rocket.

Rocket Travel. A rocket of total mass 3180 kg is traveling in outer space with a velocity of 115 m/s. To alter its course by 35.0°, its rockets can be fired briefly in a direction perpendicular to its original motion. If the rocket gases are expelled at a speed of 1750 m/s, how much mass must be expelled?

A system of particles is known to

have a total momentum of zero.

Does it necessarily follow that the

total kinetic energy of the system

is also zero?

A) yes

B) no

Conceptual QuizConceptual Quiz

A system of particles is known

to have a total momentum of

zero. Does it necessarily follow

that the total kinetic energy of

the system is also zero?

A) yes

B) no

Momentum is a vector, so the fact that ptot = 0 does

not mean that the particles are at rest! They could be

moving such that their momenta cancel out when you

add up all of the vectors. In that case, because they

are moving, the particles would have non-zero KE.



F F light heavy

A) the heavier one

B) the lighter one

C) both the same

Two boxes, one heavier than the

other, are initially at rest on a

horizontal frictionless surface.

The same constant force F acts

on each one for exactly 1 second.

Which box has more momentum

after the force acts ?


F F light heavy

av tp

F We know:

so impulse pp = = FFav av t.t.In this case FF and tt are the samesame for both boxes!

Both boxes will have the same final momentumsame final momentum.

A) the heavier one

B) the lighter one

C) both the same

Two boxes, one heavier than the

other, are initially at rest on a

horizontal frictionless surface.

The same constant force F acts

on each one for exactly 1 second.

Which box has more momentum

after the force acts ?

,

In the previous question,

which box has the larger

velocity after the force acts?

A) the heavier one

B) the lighter one

C) both the same


In the previous question,

which box has the larger

velocity after the force acts?

A) the heavier one

B) the lighter one

C) both the same

The force is related to the acceleration by Newton’s

Second Law (F = ma). The lighter box therefore has

the greater acceleration and will reach a higher

speed after the 1-second time interval.


Follow-up:Follow-up: Which box has gone a larger distance after the force acts? Which box has gone a larger distance after the force acts?

Follow-up:Follow-up: Which box has gained more KE after the force acts? Which box has gained more KE after the force acts?


p

p

A) the bowling ballA) the bowling ball

B) same time for bothB) same time for both

C) the Ping-Pong ballC) the Ping-Pong ball

D) impossible to sayD) impossible to say

A bowling ball and a Ping-Pong ball

are rolling toward you with the

same momentum. If you exert the

same force to stop each one, which

takes a longer time to bring to

rest?


We know:

Here, FF and pp are the samesame for both balls!

It will take the same amount of timesame amount of time to stop them. p

p so pp = = FFav av tt


B) same time for bothB) same time for both



A bowling ball and a Ping-Pong ball

are rolling toward you with the


same force to stop each one, which

takes a longer time to bring to

rest?

av tp

F


p

p

A bowling ball and a Ping-Pong

ball are rolling toward you with the


same force to stop each one, for

which is the stopping distancestopping distance

greater?


B) same distance for bothB) same distance for both




p

p

Use the work-energy theorem: W = W = KEKE.

The ball with less massless mass has the greater greater

speedspeed (why?)(why?), and thus the greater KEgreater KE (why(why

again?)again?). In order to remove that KE, work

must be done, where W = W = FdFd. Because the

force is the samesame in both cases, the

distance needed to stop the less massive less massive

ballball must be biggerbigger.

A bowling ball and a Ping-Pong

ball are rolling toward you with the


same force to stop each one, for

which is the stopping distancestopping distance

greater?


B) same distance for bothB) same distance for both



Amy (150 lbs) and Gwen (50 lbs) are

standing on slippery ice and push off

each other. If Amy slides at 6 m/s, what

speed does Gwen have?

A) 2 m/sA) 2 m/s

B) 6 m/sB) 6 m/s

C) 9 m/sC) 9 m/s

D) 12 m/sD) 12 m/s

E) 18 m/sE) 18 m/s

150 lbs150 lbs 50 lbs50 lbs


The initial momentum is zeroinitial momentum is zero, so the momenta of Amy and Gwen must be equal and equal and

oppositeopposite. Because p = mv, then if Amy has three times Amy has three times

more massmore mass, we see that

Gwen must have three times Gwen must have three times

more speedmore speed.

Amy (150 lbs) and Gwen (50 lbs) are

standing on slippery ice and push off

each other. If Amy slides at 6 m/s, what

speed does Gwen have?

A) 2 m/sA) 2 m/s

B) 6 m/sB) 6 m/s

C) 9 m/sC) 9 m/s

D) 12 m/sD) 12 m/s

E) 18 m/sE) 18 m/s

150 lbs150 lbs 50 lbs50 lbs


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Linear Momentum Lecturer: Professor Stephen T. Thornton