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