Momentum and Collision

1
Momentum and Collision

• Chapter 6

2
Momentum and Impulse
3
Linear Momentum

• Momentum is defined as mass times velocity.
• Momentum is represented by the symbol p, and is a
  vector quantity.
• p mv
• momentum mass ? velocity

4
Momentum

• Momentum has the dimensions of mass x length/time
  (kg x m/s)
• When do you use the word momentum?
• gaining momentum or picking up speed
• Coasting on a bike going down a hill
• You accelerate and velocity increases with time.

5
Bowling ball or basketball?

• Picture two lanes at a bowling alley.
• One with a bowling ball the other with a
  basketball going at the same speed.
• Which will exert more force on the pins?
• Why?
• More momentum

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Example

• A 2250 kg pickup truck has a velocity of 25 m/s
  to the east. What is the momentum of the truck?

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Solution

• p mv
• (2250 kg)(25 m/s east)
• 5.6 x 104 kg x m/s to the east

8
Your Turn

• A deer with a mass of 146 kg is running head-on
  toward you at a speed of 17 m/s. You are going
  north find the momentum of the deer.
• A 21 kg child on a 5.9 kg bike is riding with a
  velocity of 4.5 m/s to the northwest.
• What is the total momentum of the child and the
  bike together?
• What is the momentum of the child?
• What is the momentum of the bike?

9
Linear Momentum

• Impulse
• The product of the force and the time over which
  the force acts on an object is called impulse.
• The impulse-momentum theorem states that when a
  net force is applied to an object over a certain
  time interval, the force will cause a change in
  the objects momentum.
• F?t ?p mvf mvi
• force ? time interval change in momentum

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Linear Momentum

• Stopping times and distances depend on the
  impulse-momentum theorem.
• Force is reduced when the time interval of an
  impact is increased.

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Example

• A 1400 kg car moving westward with a velocity of
  15 m/s collides with a utility pole and is
  brought to rest in 0.30 s. Find the force
  exerted on the car during the collision.

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Solution

• F?t ?p mvf mvi
• F mvf mvi
• ?t
• F (1400kg)(0 m/s) (1400 kg)(-15 m/s)
• 0.30 s
• F 7.0 x 104 N to the east

13
Your Turn II

• A 0.50 kg football is thrown with a velocity of
  15 m/s to the right. A stationary receiver
  catches the ball and brings it to rest in 0.020
  s. What is the force exerted on the ball by the
  receiver?
• An 82 kg man drops from rest on a diving board
  3.0 m above the surface of the water and comes to
  rest 0.55 s after reacing the water. What is the
  net force on the diver as he is brought to rest?
• A 0.40 kg soccer ball approaches a player
  horizontally with a velocity of 18 m/s to the
  north. The player strikes the ball and cause it
  to move in the opposite direction with a velocity
  of 22 m/s. What is the impulse delivered to the
  ball by the player?

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Impulse-Momentum Theorem

• Stopping times and distances depend on the
  impulse-momentum theorem.
• Highway safety engineers use the impulse-momentum
  theorem to determine stopping distances and safe
  following distances for cars and trucks.

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Example

• A 2240 kg car traveling to the west slows down
  uniformly from 20.0 m/s to 5.00 m/s. How long
  does it take the car to decelerate if the force
  on the car is 8410 N to the east? How far does
  the car travel during deceleration?

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Solution

• Givens
• Mass 2240 kg
• vi 20.0 m/s to the west - 20.0 m/s
• vf 5.00 m/s to the west - 5.00 m/s
• F 8410 N to the east
• Equation to use F?t ?p
• ?t ?p / F
• ?t mvf mvi / F

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Plug and Chug

• ?t mvf mvi / F
• ?t (2240 kg)(-5.00 m/s)-(2240 kg)(-20.0 m/s)
• 8410 kg x m/s2
• ?t 4.00 s
• ?x ½ (vi vf)?t
• ?x ½ (-20.0 m/s 5.00 m/s)(4.00 s)
• ?x - 50.0 m 50.0 m to the west

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Your Turn III

• A 2500 kg car traveling to the north is slowed
  down uniformly from an initial velocity of 20.0
  m/s by a 6250 N braking force acting opposite the
  cars motion. Use the impulse-momentum theorem
  to answer the following questions
• What is the cars velocity after 2.50 s?
• How far does the car move during 2.50 s?
• How long does it take the car to come to a
  complete stop?

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Impulse-Momentum Theorem

• Force is reduced when the time interval of an
  impact is increased.
• Examples
• Nets or giant air mattresses used to catch
  people.
• Page 203 in your book shows an image of a girl
  being tossed in the ir and caught in a blanket.
• The blanket gives way and extends the time of
  collision to change the momentum over a longer
  period of time.
• Consider the egg on the next slide and explaing
  what is happening

20
Impulse-Momentum Theorem
21
PNBW

• Page 204
• Physics 1-3
• Honors 1-5

22
Conservation of Momentum
23
Momentum is Conserved

• So far we only have considered the momentum of
  only one object at a time.
• Now we will look at two or more objects
  interacting with each other.
• Picture this. . .
• You are playing pool. You strike the cue ball it
  hits the 8 ball. The 8 ball had no momentum
  before they collided.
• During the collision the cue ball loses momentum
  and the 8 ball gains momentum.
• The momentum the cue ball loses is the same
  amount that the 8 ball gained.

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Momentum is Conserved

• The Law of Conservation of Momentum
• The total momentum of all objects interacting
  with one another remains constant regardless of
  the nature of the forces between the objects.
• m1v1,i m2v2,i m1v1,f m2v2,f
• total initial momentum total final momentum

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Momentum is Conserved

• The total momentum of all objects interacting
  with one another remains constant regardless of
  the nature of the forces between the objects.
• Go back to the pool table example. The cue ball
  and the 8 ball do not have a constant momentum,
  but the total momentum is constant.

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Momentum is Conserved

• Consider objects pushing away from each other.
• You jump up. You push of the Earth. Take you
  mv.
• Lets say 60 kg m/s upward. That means that the
  Earth must have a corresponding momentum of 60 kg
  m/s downward. However, the has an enormous mass
  which means its velocity is very small.

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Momentum is Conserved

• Picture this . . .
• Two people on skates facing one another. They
  push away from one another. Initially, they are
  both at rest with a momentum of 0. When the push
  away, they move in opposite directions with equal
  but opposite momentum, so that the total momentum
  is 0.

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Example

• A 76 kg boater, initially at rest in a stationary
  45 kg boat, steps out of the boat and onto the
  dock. If the boater moves out of the boat with a
  velocity of 2.5 m/s to the right, what is the
  final velocity of the boat?

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Solution

• Given
• m1 76 kg m2 45 kg
• v1,i 0 v2,i 0
• v1,f 2.5 m/s to the right
• Unknown
• v2,f ?

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Solution

• Choose an equation or situation Because the
  total momentum of an isolated system remains
  constant, the total initial momentum of the
  boater and the boat will be equal to the total
  final momentum of the boater and the boat.
• m1v1,i m2v2,i m1v1,f m2v2,f

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Solution

• Because the boater and the boat are initially at
  rest, the total initial momentum of the system is
  equal to zero. Therefore, the final momentum of
  the system must also be equal to zero.
• m1v1,f m2v2,f 0
• Rearrange the equation to solve for the final
  velocity of the boat.

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Solution

• Substitute the values into the equation and
  solve

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Solution

• The negative sign for v2,f indicates that the
  boat is moving to the left, in the direction
  opposite the motion of the boater. Therefore,

v2,f 4.2 m/s to the left
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Your Turn IV

• A 63.0 kg astronaut is on a space walk when the
  tether line to the shuttle breaks. The astronaut
  is able to throw a spare 10.0 kg oxygen tank in a
  direction away from the shuttle with a speed of
  12.0 m/s, propelling the astronaut back to the
  shuttle. Assuming that the astronaut starts from
  rest with respect to the shuttle, find the
  astronauts final speed with respect to the
  shuttle after the tank is thrown.
• An 85.0 kg fisherman jumps from a dock into a
  135.0 kg rowboat at rest on the west side of the
  dock. If the velocity of the fisherman is 4.30
  m/s to the west as he leaves the dock, what is
  the final velocity of the fisherman and the boat?

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Your Turn IV

• Each croquet ball in a set has a mass of 0.50 kg.
  The green ball, traveling at 12.0 m/s, strikes
  the red ball, which is at rest. Assuming that
  the croquet ball slide on a frictionless surface
  and all collisions are head-on, find the final
  speed of the red ball in each of the following
  situations
• The green ball stops moving after it strikes the
  red ball.
• The green ball continues moving after the
  collision at 2.4 m/s in the same direction.

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Momentum is Conserved

• Newtons third law leads to conservation of
  momentum
• During the collision, the force exerted on each
  bumper car causes a change in momentum for each
  car.
• The total momentum is the same before and after
  the collision.

37
PNBW

• Page 211
• Physics 1-3
• Honors 1-4

38
Elastic and Inelastic Collisions
39
Collisions

• Perfectly inelastic collision
• A collision in which two objects stick together
  after colliding and move together as one mass is
  called a perfectly inelastic collision.
• Example The collision between two football
  players during a tackle.
• Conservation of momentum for a perfectly
  inelastic collision
• m1v1,i m2v2,i (m1 m2)vf
• total initial momentum total final momentum

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Example

• A 1850 kg luxury sedan stopped at a traffic light
  is struck from behind by a compact car with a
  mass of 975 kg. The two cars become entangled as
  a result of the collision. If the compact car
  was moving with a velocity of 22.0 m/s to the
  north before the collision, what is the velocity
  of the entangled mass after the collision?
• Given m1 1850 kg m2 975 kg
• v1,i 0 m/s v2,i 22.0 m/s north
• Unknown vf

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Solution

• Choose your equation
• m1v1,i m2v2,i (m1 m2)vf
• vf m1v1,i m2v2,i
• (m1 m2)
• vf (1850 kg)(0 m/s) (975 kg)(22.0 m/s)
• (1850 kg 975 kg)
• vf 7.59 m/s north

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Your Turn V

• A 1500 kg car traveling at 15.0 m/s to the south
  collides with a 4500 kg truck that is initially
  at rest at a stoplight. The car and truck stick
  together and move together after the collision.
  What is the final velocity of the two-vehicle
  mass?
• You are shopping at Publix and toss a 9.0 kg bag
  of rice into a stationary 18.0 kg shopping cart.
  The bag hits the cart with a horizontal speed of
  5.5 m/s toward the front of the cart. What is
  the final speed of the cart and the bag?
• A 47.4 kg student runs down the sidewalk and
  jumps with a horizontal velocity of of 4.20 m/s
  onto a stationary skateboard. The student and the
  skateboard move down the sidewalk with a speed of
  3.95 m/s. Find the following
• The mass of the skatboard
• How fast the student would have to jump to have a
  final speed of 5.00 m/s

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Kinetic Energy in Inelastic Collisions

• In an inelastic collision the total kinetic
  energy does not remain constant when the objects
  collide and stick together.
• Some energy is converted into sound energy and
  internal energy as the objects deform during the
  collision.
• Elastic in physics refers to a material that when
  work is done to deform the material during a
  collision the same amount of work is done to
  return the material to its original shape.
• Inelastic material does not return to its
  original shape and therefore some energy is
  converted to sound or heat.

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Example

• Two clay balls collide head-on in a perfectly
  inelastic collision. The first ball has a mass of
  0.500 kg and an initial velocity of 4.00 m/s to
  the right. The second ball has a mass of 0.250 kg
  and an initial velocity of 3.00 m/s to the left.
  What is the decrease in kinetic energy during the
  collision?

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Solution

• Given m1 0.500 kg m2 0.250 kg
• v1,i 4.00 m/s to the right
• v1,i 4.00 m/s
• v2,i 3.00 m/s to the left
• v2,i 3.00 m/s
• Unknown ?KE ?

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Solution

• The change in kinetic energy is simply the
  initial kinetic energy subtracted from the final
  kinetic energy.
• ?KE KEi KEf
• Determine both the initial and final kinetic
  energy.

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Solution

• Use the equation for a perfectly inelastic
  collision to calculate the final velocity.

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Solution

• Next calculate the initial and final kinetic
  energy.

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Solution

• Finally, calculate the change in kinetic energy.

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Your Turn VI

• A 0.25 kg arrow with a velocity of 12 m/s to the
  west strikes and pierces the center of a 6.8 kg
  target. What is the final velocity of the
  combined mass? What is the decrease in kinetic
  energy during the collision?
• A clay ball with a mass of 0.35 kg hits another
  0.35 kg ball at rest, and the two stick together.
  The first ball has an initial speed of 4.2 m/s.
  What is the final speed of both balls? Calculate
  the decrease in kinetic energy that occurs during
  the collision.
• A 56 kg ice skater traveling at 4.0 m/s to the
  north meets and joins hands with a 65 kg skater
  traveling at 12.0 m/s in the opposite direction.
  Without rotating the two skaters continue skating
  together. What is the final velocity of the two
  skaters and what is the decrease in kinetic
  energy durning the collision?

51
Elastic Collisions

• A collision in which the total momentum and the
  total kinetic energy are conserved is called an
  elastic collision.
• Elastic means that after a collision the objects
  remain separated.
• Two objects collide and return to their original
  shapes with no loss of total kinetic energy.
  After the collision the two objects move
  separately.
• Both the total momentum and total kinetic energy
  are conserved.

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Real Collisions

• Most collisions are not perfectly inelastic (they
  dont stick together and move as one)
• Most collisions are not elastic.
• Even nearly elastic collisions result in some
  decrease of kinetic energy.
• A football deforms when kicked
• A sound is produced (sound signifies a decrease
  in kinetic energy)

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For all intensive purposes

• We will consider all collisions in which objects
  do not stick together to be elastic collisions.
• Therefore, total momentum and total kinetic
  energy will stay the same before and after the
  collision.

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Kinetic Energy is Conserved in Elastic Collisions
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Elastic Collisions

• The total momentum is always constant throughout
  the collision. In addition, if the collision is
  perfectly elastic, the value of the total kinetic
  energy after the collision is equal to the value
  before the collision.

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Example

• A 0.015 kg marble moving to the right at 0.225
  m/s makes an elastic head-on collision with a
  0.030 kg shooter marble moving to the left at
  0.180 m/s. After the collision, the smaller
  marble moves to the left at 0.315 m/s. Assume
  that neither marble rotates before or after the
  collision and that both marbles are moving on a
  frictionless surface. What is the velocity of
  the 0.030 kg marble after the collision?

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Solution

• Given m1 0.015 kg m2 0.030 kg
• v1,i 0.225 m/s to the right, v1,i 0.225
  m/s
• v2,i 0.180 m/s to the left, v2,i 0.180 m/s
• v1,f 0.315 m/s to the left, v1,i 0.315 m/s
• Unknown
• v2,f ?

58
Solution
59
Solution
60
Your Turn VII

• Two billiard balls, each with a mass of 0.35 kg,
  strike each other head-on. One ball is initially
  moving left at 4.1 m/s and ends up moving right
  at 3.5 m/s. The second ball is initially moving
  to the right at 3.5 m/s. Assume that neither
  ball rotates before or after the collision and
  that both balls are moving on a frictionless
  surface. Predict the final velocity of the
  second ball.
• A 0.015 kg marble sliding to the right at 12.5
  m/s on a frictionless surface makes an elastic
  head-on collision with a 0.015 kg marble moving
  to the left at 18.0 m/s. After the collision,
  the first marble moves to the left at 18.0 m/s.
  Find the velocity of the second marble after the
  collision. Verify your answer by calculating the
  total kinetic energy before and after the
  collision.

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Your Turn VII (continued)

• A 16.0 kg canoe moving to the left at 12.5 m/s
  makes an elastic head-on collision with a 14.0 kg
  raft moving to the right at 16.0 m/s. After the
  collision, the raft moves to the left at 14.4
  m/s. Disregard and effects of the water. Find
  the velocity of the canoe after the collision.
  Calculate the total kinetic energy before and
  after the collision.

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Types of Collisions
63
PNBW

• Pg 220
• Physics 1-4
• Honors 1-5