Momentum and Impulse

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Momentum and Impulse
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Flow Chart for Problem Solving

• Can the problem be solved with an identification
  of terms and the application of one formula? For
  example, given the mass and velocity of an
  object, determine the momentum. Click here.
• Can the problem be solved with an identification
  of terms and the application of several formulas?
  Click here.

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Knowledge Level Problem

• What is the momentum of a 2000 kg truck traveling
  at 20 m/s?
• Label the knowns and unknown m 2000 kg v 20
  m/s, p ?
• Identify connecting relationship p mv
• Rearrange, if necessary, to solve for unknown p
  mv.
• Insert knowns and finish p 40,000 kgm/s

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• Is there a fundamental concept involved or just
  the application of several knowledge level
  relationships?
• For example, a mass, m, is dropped from a height,
  h. What is the momentum of the object at impact?
  This is a problem using the knowledge that p
  mv and kinematics formula, v2 vo2 2 a x.
  Click here.
• Is there a principle involved? Example
  Conservation of momentum or energy?
• Click here.

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• A ball, of mass 0.10 kg, is dropped from a height
  of 12 m. What is its momentum when it strikes
  the ground, in kgm/s?
• Label knowns m 0.1 kg, h 12 m.
• Unknown p ?.
• Relationships p mv, v2 vo2 2ax
• Combine to solve for unknown
• p m(2gh)1/2 (0.1)(29.812)1/2 1.5 kgm/s

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Principles and Concepts

• Some problems involve one object striking another
  object. For example, a baseball bat striking a
  baseball. Click here.
• Some problems will use conservation principles of
  momentum and energy to relate the situation
  before and after a collision. Click here.

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• A constant 6.0-N net force acts for 4.0 s on a 12
  kg object. What is the object's change of
  velocity?
• Knowns F 6 N, ?t 4 s, m 12 kg
• Unknown ?v ?
• Relationship Impulse-Momentum Theorem, F ?t
  ?p
• m?v
• Rearrange for unknown ?v F?t/m

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Conservation of Momentum

• The first step is to calculate the momentum of
  each object before collision.
• Remember these are vectors.
• Two identical 1500-kg cars are moving
  perpendicular to each other. One moves with a
  speed of 25 m/s due north and the other moves at
  15 m/s due east. What is the total momentum of
  the system?
• P(total) P(car 1) P(car 2), where the Ps are
  vectors!

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• P(car 1)1500 kg25 m/s j37,500 kgm/s j
• P(car 2)1500 kg15 m/s i22,500 kgm/s i.
• P(total) (22,500 i 37,500 j )kgm/s
• Magnitude of
• P(total)v(22,5002 37,5002)
• 43732 kgm/s at an angle of
• tanT 37,500/22,500
• T 59o

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• If a collision occurs we need to know what type
  of collision it is, i.e. elastic, inelastic or
  partially elastic.
• Elastic
• Inelastic
• Partially elastic

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Elastic Collision

• Momentum is conserved.
• Kinetic Energy is conserved.
• A 50-gram ball moving 10 m/s collides head-on
  with a stationary ball of mass 100 g. The
  collision is elastic. What is the speed of each
  ball immediately after the collision?
• Knowns m1 0.05 kg, v01 10 m/s i, m2 0.1
  kg, v02 0 m/s

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• Calculate total momentum prior to collision p
  0.05 10 0.5 kgm/s i
• Write equation for momentum after collision, p
  m1v1 m2v2
• 0.05v1 0.1v2
• Conservation of momentum gives
• 0.5 0.05v1 0.1v2
• So, v2 5 - 0.5 v1
• We need another relationship between the
  velocities before and after impact. Conservation
  of kinetic energy applies!

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• K(before) K(after)
• 1/2m1v102 1/2m2v202 1/2m1v12 1/2m2v22
• 0.05102 0 0.05v12 0.1v22
• 5 0.05v12
  0.1v22
• v22 50 -
  0.5v12
• Now you have two formulas for the same variables
  which can be solved by substitution.

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Inelastic Collisions

• Momentum is always conserved.
• Kinetic energy is not conserved.
• ExampleA railroad freight car, mass 15,000 kg,
  is allowed to coast along a level track at a
  speed of 2.0 m/s. It collides and couples with a
  50,000-kg second car, initially at rest and with
  brakes released. What is the speed of the two
  cars after coupling?

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• Determine momentum before collision
• P 15,000 kg 2 m/s 50,000 kg 0 m/s
• P(before) 30,000 kg m/s
• Momentum after collision
• P(after) (15,000 50,000) V
• 35,000 V
• Conservation of momentum
• P(before) P(after)
• 30,000 65,000 V
• V 0.46 m/s

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Partially Elastic

• Momentum is conserved.
• You need to know three of the four velocities
  involved in the problem. For instance, you need
  to know how fast the objects are moving prior to
  collision, and at least one of the objects speed
  after collision.

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• A 10 g bullet moving horizontally at 375 m/s
  penetrates a 3 kg wood block resting on a
  frictionless horizontal surface.  If the bullet
  slows down to 300 m/s after emerging from the
  block, what is the speed of the block immediately
  after the bullet emerges.
• Knowns M1 0.01 kg, v10 375 m/s, M2 3 kg,
  v20 0 m/s, v1 300 m/s,
• Unknown v2 ?
• P(before) 0.01 375 3.75 kgm/s
• P(after) 0.01300 3(v2) 3 3v2
• Conserve momentum 3.75 3 3v2