Momentum

The secret of collisions and explosions

Who Pushes Who

• Rin Tin Tin and the Refrigerator meet at the 50 yard line

Mass 20 Kg

Speed 17 m/s

Mass 160 Kg

Speed 2 m/s

Who pushes who over the 50 yard line?

Momentum = Mass x Velocity

Whoever has the most momentum = mv wins

Mass 20 Kg

Speed 17 m/s

Mass 160 Kg

Speed 2 m/s

Momentum = 340 Kg m/s Momentum = 320 Kg m/s

Momentum = mv

• Mass times velocity!

• Velocity is a vector but here we can usually think of it as speed

• Example: A speeding car has a mass of 1000 Kg and a speed of 20 m/s. What is its momentum?

• mv = 1000kg x 20 m/s = 20,000 kg m/s

Unit of Momentum

Kg m/s

No nickname like “Newton;” maybe it could have been called an “Isaac.”

Jump to conservation

Momentum is a Vector

• p = mv• Force is required to change the momentum of

an object. Newton stated his 2nd law:

F = p/t The rate of change of momentum of a body

equals the net force exerted on it Equivalent to F = ma

Proof of equivalence of two forms of Newton’s 2nd Law

F = p/t = (mv –mv0)/t =

m(v - v0)/ t = mv/t = maQ: Which form of the law is more general? (which includes the possibility that the mass could change?)

Example

• Water leaves a hose at a rate of 1.5 kg/s with speed 20 m/s and hits a car without splashing back. What force is exerted by the water on the car?

F = p/t = (p final - pinitial )/ t = (0 – 30kg m/s)/1sec = -30N

Momentum is Conserved

• The total momentum of an isolated system of bodies remains the same

• “isolated” means net external force is zero• Momentum before = momentum after• m1v1 + m2v2 = m1v1’ + m2v2’• Applies to all interactions, especially

collisions, explosion-like events, and “dumpings”

• Closely related to Newton’s 3rd law

Law of Conservation of Momentum

• Momentum before = momentum after

• m1v1 + m2v2 = m1v1’ + m2v2’

Apostrophe thingee is pronounced “prime”

Two Kinds of Collision

• Inelastic - sticking Examples: glue balls fly into each other,air track

gliders with clay

• Elastic – bouncing• Examples: hard balls or protons collide

Courtesy St. Mary College Physics Department

Courtesy Deer Vally HS Espace Academy

Two Kinds of Collision(video)

• Inelastic - sticking Examples: glue balls fly

into each other,air track gliders with clay

• Elastic – bouncing• Examples: hard balls or

protons collide

Courtesy St. Mary College Physics Department

Courtesy Stephanie Wong

Video Full Screen

New Vocabulary

• Momentum

• Conserved

• Isolated

• Elastic collision

• Inelastic collision

Mass times velocity

Stays the same

By itself, net force on it is zero

Objects bounce

Objects stick

Objects moving toward each other

• If two carts of equal mass approach each other with equal speed what is the total momentum before collision?

Answer: zero

Dumping Example

• A 10 kg cart rolls on a frictionless track at speed v. Suddenly 10 kg of rocks are dropped straight down into the cart. What happens to its speed? How come?

Answer v/2 ; momentum mv is conserved, m doubled so v must go in half.

Courtesy Easyhaul Cart Inc.

Explain This

• A rock falls to earth. Is momentum conserved? Include earth in your explanation

• Does the earth really come up to meet the rock?

You Predict

• Glider moving with speed v hits glider of equal mass at rest. They stick. What will be speed of stuck together gliders after the collision?

Answer: v/2

Railroad Cars CollideInelastically(stick)

• A 10,000 kg railcar moving 2.4 m/s hits and sticks with an identical car at rest. What is the final speed of the two cars?

m1v1 + m2v2 = (10,000 kg) (2.4 m/s) + 0 =

2.4 x 10 4 kg m/s = (m1 + m2)v’ v’= 2.4 x 104 kg m/s /2.0 x 10 4 kg = 1.2

m/s

New example

• A 3 kg glider moves to the right at 4.0 m/s and collides inelastically with a 1 kg glider at rest. What is the final speed of the two joined together gliders?

• 12 kg m/s = 4kg x vf

• vf = 3 m/s

Next example

• A 3 kg glider moves to the right at 4.0 m/s and collides inelastically with a 1 kg glider moving to the right at 1.0 m/s. What is the final speed of the two joined together gliders?

• 12 kg m/s + 1kg x 1 m/s = 13 kg m/s = 4.0 kg vf

• vf = 3.25 m/s

Next example

• A 3 kg glider moves to the right at 4.0 m/s and collides inelastically with a 1 kg glider moving to the right at 2.0 m/s. What is the final speed of the two joined together gliders?

• 12 kg m/s + 1kg x 2 m/s = 14 kg m/s = 4.0 kg vf

• vf = 3.50 m/s

Next example

• A 3 kg glider moves to the right at 4.0 m/s and collides inelastically with a 1 kg glider moving to the left at 1.0 m/s. What is the final speed of the two joined together gliders?

• 12 kg m/s - 1kg x 1 m/s = 11 kg m/s = 4.0 kg vf

• vf = 2.75 m/s

Next example

• A 3 kg glider moves to the right at 4.0 m/s and collides inelastically with a 1 kg glider moving to the left at 5.0 m/s. What is the final speed of the two joined together gliders?

• 12 kg m/s - 1kg x 5 m/s = 7 kg m/s = 4.0 kg vf

• vf = 1.75 m/s

Next example

• A 3 kg glider moves to the right at 4.0 m/s and collides inelastically with a 1 kg glider moving to the left at 20.0 m/s. What is the final velocity of the two joined together gliders?

• 12 kg m/s - 1kg x 20 m/s = -8 kg m/s = 4.0 kg vf

• vf = -2.00 m/s

Sled Collision

• Kids on a sled, total mass 100kg move to the right at 4.0 m/s. They collide inelastically with other kids, mass 150 kg, moving to the left at 2.5 m/s. Find the final velocity of the two sleds.

solution

• m1v1 + m2v2 = m1v1’ + m2v2’

• 100Kg x 4m/s –150Kg x 2.5m/s =• 400 Kg m/s – 375 Kg m/s = 25 Kg m/s =

• 250 Kg x vf

• Vf = 0.1 m/s

Sled Collision in reverse• Kids on a sled, total mass 100kg move

to the right at unknown speed v. They collide inelastically with other kids, mass 150 kg, moving to the left at 2.5 m/s. The final velocity of the two sleds is 0.2 m/s. Find v

V = 4.25 m/s

Explosion like Event

• If a stationary student on a skateboard throws a rock with momentum 10 kg m/s, what momentum will the student get?

Answer; -10 kg m/s

Recoil of a pistol(explosion-like event)

• What is the recoil velocity of a 1 kg pistol that shoots a .02 kg bullet at 400 m/s?

• Initial momentum = 0 = mBvB+mpvp = (0.02kg x 400 m/s) + 1kg x vp

vp = - 8 m/s Q: Does the shooter recoil too?

Skip think and explain # 8

Think and Explain

8) m1v1 + m2v2 = m1v1’ + m2v2’

Let mass of flat car = m Then mass of diesel engine = 4m

V1 = 5 km/h v2 = 0

4m * 5 km/h + 0 = 5 m Vf

20 m = 5m Vf

Vf = 4 km/h

9) A) 5 kg * 1 m/s + 0 = 6 kg Vf

Vf = 5/6 m/s

B) 5 kg * 1 m/s – 1 kg * 4 m/s = 6 kg * Vf

5 kg * m/s – 4 kg * m/s = 6 kg * Vf

1 Kg * m/s = 6kg Vf

Vf=1/6 m/s

“Impulse” Why should you

• Bend your knees when you land?

• Pull back when the baseball enters

• your mitt?

• Follow through when you swing?

• Not walk into a punch? (like Mike Tyson did)

Impulse

• Impulse = F t = p

• Impulse is product of force and time during which force acts

• Impulse equals change of momentum

• F is usually non uniform and time interval is usually short

You Predict

• Two gliders of equal mass collide elastically. The first is moving with speed v. The second is at rest. What happens?

First one stops, second moves off at speed v

But why? Find the answer yourself and get extra credit

Energy Conservation in Elastic and Inelastic Collisions• Elastic – kinetic energy is conserved as

well as momentum and total energy

• Inelastic – kinetic energy is not conserved – some energy turns into heat

• Elastic – bounce

• Completely inelastic - stick

Ballistic Pendulum

• A bullet of mass m is fired into a block of wood of mass M suspended from a string. The bullet remains in the block which rises a height h. What was the speed of the bullet? Show that

• v = (2gh)1/2(m + M)/m

gh2

h

Collisions in Two Dimensions

• Remember momentum p is a vector

• x and y components are conserved separately

• What is the total vertical momentum?

1

2