Unit 5: Conservation of Momentum

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Unit 5 Conservation of Momentum

• Force and Momentum and Conservation of Momentum
  (9-1,9-2, 9-3)
• Collisions (9-4, 9-5, 9-6)
• Collisions, Center of Mass (9-7,9-8)
• Catch-up, Rocket Propulsion, and Quiz Three
  Review (9-9,9-10)

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Schedule

• No class Friday, March 30th!
• Next test Friday, April 6th
• Please, work on your problems sets and extra
  credit.
• If you need information to compute your grade
  send email or see me at my office!

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Conservation Laws

• There are many conservation laws, weve already
  discussed Conservation of Energy.
• In this course we will also discuss
• Conservation of Momentum
• Conservation of Angular Momentum
• Many others charge, baryon number, lepton
  numberall are a consequence of some fundamental
  symmetry of nature.
• When we combine the conservation of energy,
  momentum, and angular momentum we can beautifully
  describe complex systems of objects.

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Momentum

• Linear momentum is a deceptively simple quantity
  equal to the product of an objects mass, m, and
  velocity, v
• pmv
• Properties of momentum
• A vector with same direction as the velocity,
  which requires a reference frame
• Magnitude equal to mv, increases linearly with m
  and with velocity.
• SI unit is kg-m/s but carries no explicit name.
• Newton actually did have a name he called it the
  quantity of motion. (Easy to see why it didnt
  stick!)

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Force and Momentum

• A force is required to change magnitude or
  direction of momentum with respect to time.
• Actually this is similar to Newtons original
  statement of the 2nd Law The rate of change of
  momentum of an object is equal to the net force
  applied to it.
• Or in symbols

• A quick derivation shows the two versions of the
  2nd law are equivalent
• Note we assumed that the mass was constant!
• Actually the equality of the force to the change
  of momentum is more general and useful. For
  instance, if we allow the mass to change we can
  describe propulsion

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Example Water Hitting a Car

• Water leaves a hose and hits a car at a rate of
  1.5 kg/s and a speed of 20 m/s.
• What is the force exerted by the water on the
  car?
• What if the water splashes off at -5m/s? Is the
  force greater or less?

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Experimental Conservation of Momentum

• Consider head-on collision of two hard balls
• Assume no net external forces.
• In the 17th century and predating Newton it was
  found that
• Individual momentum can change
• Vector sum of momentum was observed to be
  constant.

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Theoretical Conservation of Momentum

• The experimental result can be explained using
  Newtons Laws.
• Consider two colliding objects
• Initial momentum p1 and p2
• Final momentum p1 and p2
• Object 1 exerts a force F on object 2
• Object 2 exerts force F on object 1.
• No other significant forces involved.

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Example Goal Line Stand

• A 90-kg fullback attempts to dive over the goal
  line with a velocity of 6.00 m/s. He is met at
  the goal line by a 110-kg linebacker moving at
  4.00 m/s in the opposite direction.  The
  linebacker holds on to the fullback. 
• Does the fullback cross the goal line?

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Example The Kick of a Rifle

• What is the recoil velocity of a 5.0-kg rifle
  that shoots at 0.050-kg bullet at a muzzle
  velocity of 120m/s?
• What would be the initial velocity of a 75-kg
  rifleperson?

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Example Billiard Ball Collision in 2-D

• A billiard ball moving at 3.0m/s in the x
  direction strikes a ball of equal mass initially
  at rest. The two balls move off at 45o wrt to
  the x axis as shown.
• What are the speeds of the two balls after the
  collision?

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Generalization to a System of Objects

• The derivation can be extended to any number of
  objects.
• The total initial momentum is
• P m1v1m2v2mnvnSmivi Spi
• Which can be differentiated with respect to time
• dP/dt d(Spi)/dt SFi
• Where Fi is the net force on the ith object.
  There are internal external forces. But
  remember that internal forces come in equal and
  opposite pairs so they will cancel in the
  summation. Thus dP/dt SFexternal
• Now if the net external forces are zero then the
  change in momentum of the system is zero and
  Pconstant!

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• The Law of Conservation of Momentum
• The total momentum of an isolated system of
  bodies remains constant.

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Example An Exploded Firecracker

• A firecracker with a mass of 100g, initially at
  rest, explodes into 3 parts.   One part with a
  mass of 25g moves along the x-axis at 75m/s.  One
  part with mass of 34g moves along the y-axis at
  52m/s. 
• What is the velocity of the third part? 

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Impulse

• Billiard balls interact almost instantaneously,
  certainly in a fraction of a second.
• As shown in the figures, when two objects
  interact the contact force rises rapidly from
  zero to a maximum and just as quickly falls to
  zero.
• This occurs in a small time interval Dt, during
  which there is an impulse of force.

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• Properties of Impulse
• Units are N-s or kg-m/s.
• Equal to the area under the force versus time
  curve.
• Convenient to calculate in terms of the average
  force during an event. Where the average force is
  defined as the constant force which, if acting
  over the time interval of the interaction would
  produce the same impulse and momentum change

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Example The Impulse of a Jump

• Calculate the impulse on a 70-kg person when
  landing on the ground after jumping from 3.0m.
• Estimate the average force exerted on the
  persons feet by the ground if landing is
• Stiff legged (body moves 0.01m)
• With bent knees. (body moves (0.50m)

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• Remember
• We cant get the impulse by the time integral of
  the force, but we can get it by calculating the
  momentum.
• The final velocity is zero!
• The initial velocity can be determined by using
  energy conservation

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Next

• Collisions Elastic, Inelastic, 2 and
  3Dimensional
• No class Friday, March 30th!
• Next test Friday, April 6th