Conservation of Momentum

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Conservation of Momentum

• Chris Sedlack

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Momentum Review

• The momentum of a particle depends on its mass
  and its velocity
• p mv
• The force on a particle is the change in momentum
  over time
• F dp/dt d(mv)/dt v dm/dt m dv/dt
• ma (for constant m)

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Conservation of Momentum

• When there is no net external force on a system,
• S pi S pf
• Allows us to make predictions about systems even
  without knowledge regarding internal forces

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1-D Example

• A block of mass m1 10 kg slides along a
• frictionless surface with velocity vi to the
  right
• until it collides with a block of mass m2 5 kg.
  
• After the collision, the first block continues to
  
• the right with a speed v1f 1 m/s, and the
• second block moves to the right with a speed
• v2f 4 m/s. What was the initial speed of the
• first block?

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1-D Example Solution

• Initial momentum
• pi m1v1i m2v2i m1vi
• Final momentum
• pf m1v1f m2v2f
• Conservation of momentum
• pi pf
• m1vi m1v1f m2v2f
• vi v1f (m2/m1) v2f
• Numerical result
• vi 1 m/s (5 kg/10 kg)(4 m/s)
• vi 3 m/s

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Momentum Conservation in 2-D

• Momentum is a vector
• We must conserve each component individually

Before collision px,i m1vi
py,i 0 After collision
px,f m1v1 cos ?1 m2v2 cos ?2
py,f m1v1 sin ?1 - m2v2 sin ?2 Conservation of
momentum m1vi m1v1 cos ?1 m2v2
cos ?2 0 m1v1 sin ?1 - m2v2 sin
?2
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Momentum Conservation and Newtons Third Law

• For a two-particle system
• ?p ?p1 ?p2
• ?p1 ? F1 dt ?p2 ?
  F2 dt
• ?p1 ? (F12 F1,ext) dt ?p2 ? (F21
  F2,ext) dt
• ?p ? (F12 F21 F1,ext F2,ext) dt
• No Net External Force F1,ext F2,ext 0
• Newtons Third Law F12 -F21
• ? ?p 0
• Momentum is conserved!

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Conservation of momentum Summary

• Conservation of momentum is a consequence of
  Newtons Second and Third Laws
• It applies when there is no net external force on
  a system
• It allows us to calculate properties of a system
  even when the internal forces are unknown