Linear Momentum

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Linear Momentum

• Physics
• Montwood High School

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Linear Momentum

• Linear momentum of an object is the mass of the
  object multiplied by its velocity.
  
• Momentum p mv
• Unit kgm/s or Ns
• Both momentum and kinetic energy describe the
  motion of an object and any change in mass and/or
  velocity will change both the momentum and
  kinetic energy of the object.

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Linear Momentum

• Momentum refers to inertia in motion.
• Momentum is a measure of how difficult it is to
  stop an object a measure of how much motion an
  object has.
  
• More force is needed to stop a baseball thrown at
  95 mph than to stop a baseball thrown at 45 mph,
  even though they both have the same mass.

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Linear Momentum

• More force is needed to stop a train moving at 45
  mph than to stop a car moving at 45 mph, even
  though they both have the same speed.
  
• Both mass and velocity are important factors when
  considering the force needed to change the motion
  of an object.
  

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Impulse

• Impulse (J) forcetime
• Equation J Ft Unit Ns
• The impulse of a force is equal to the change in
  momentum of the body to which the force is
  applied. This usually means a change in
  velocity.
• Ft m?v where ?v vf - vi
• The same change in momentum can be accomplished
  by a small force acting for a long time or by a
  large force acting for a short time.
  

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Impulse

• If your car runs into a brick wall and you come
  to rest along with the car, there is a
  significant change in momentum. If you are
  wearing a seat belt or if the car has an air bag,
  your change in momentum occurs over a relatively
  long time interval. If you stop because you hit
  the dashboard, your change in momentum occurs
  over a very short time interval.

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Impulse

• If a seat belt or air bag brings you to a stop
  over a time interval that is five times as long
  as required to stop when you strike the
  dashboard, then the forces involved are reduced
  to one-fifth of the dashboard values. That is
  the purpose of seat belts, air bags, and padded
  dashboards. By extending the time during which
  you come to rest, these safety devices help
  reduce the forces exerted on you.
• If you want to increase the momentum of an object
  as much as possible, you apply the greatest force
  you can for as long a time as possible.
  

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• A 1000 kg car moving at 30 m/s
• (p 30,000 kg m/s) can be stopped by 30,000 N
  of force acting for 1.0 s (a crash!)
  
• or by 3000 N of force acting for 10.0 s (normal
  stop)
  

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Impulse and Bouncing

• Impulses are greater when bouncing takes place.
• The impulse required to bring an object to a stop
  and then throw it back again is greater than the
  impulse required to bring an object to a stop.
  

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Conservation of Linear Momentum

• In a closed system of objects, linear momentum is
  conserved as the objects interact or collide.
  The total vector momentum of the system remains
  constant.
• p before interaction p after interaction

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Perfectly Inelastic Collisions

• Perfectly inelastic collisions are those in which
  the colliding objects stick together and move
  with the same velocity.
  
• Kinetic energy is lost to other forms of energy
  in an inelastic collision.

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Inelastic Collision Example

• Cart 1 and cart 2 collide and stick together
• Momentum equation
• Kinetic energy equation
• v1 and v2 velocities before collision
• v ? velocity after collision

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Directions for Velocity

• Momentum is a vector, so direction is important.
• Velocities are positive or negative to indicate
  direction.
  
• Example bounce a ball off a wall

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Inelastic Collisions

• Kinetic energy is lost when the objects are
  deformed during the collision.
  
• Momentum is conserved.

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Elastic Collisions

• Momentum and kinetic energy are conserved in an
  elastic collision.
  
• The colliding objects rebound from each other
  with NO loss of kinetic energy.

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Elastic Collision Example

• Example mass 1 and mass 2 collide and bounce
  off of each other
  
• Momentum equation
• Kinetic energy equation
• v1 and v2 velocities before collision
• v1? and v2? velocities after collision
• Velocities are or to indicate directions.

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Elastic Collisions Involving an Angle

• Momentum is conserved in both the x-direction and
  in the y-direction.
  
• Before

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Elastic Collisions Involving an Angle

• After

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Elastic Collisions Involving an Angle

• Directions for the velocities before and after
  the collision must include the positive or
  negative sign.
  
• The direction of the x-components for v1 and v2
  do not change and therefore remain positive.
  
• The directions of the y-components for v1 and v2
  do change and therefore one velocity is positive
  and the other velocity is negative.

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Elastic Collisions Involving an Angle

• px before px after
• py before py after
• Velocity after collision

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Elastic Collisions

• Perfectly elastic collisions do not have to be
  head-on.
  
• Particles can divide or break apart.
• Example nuclear decay (nucleus of an element
  emits an alpha particle and becomes a different
  element with less mass)

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Elastic Collisions

• mn mass of nucleus
• mp mass of alpha particle
• vn velocity of nucleus before event
• vn velocity of nucleus after event
• vp velocity of particle after event

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Recoil

• Recoil is the term that describes the backward
  movement of an object that has propelled another
  object forward. In the nuclear decay example,
  the vn would be the recoil velocity.

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Head-on and Glancing Collisions

• Head-on collisions occur when all of the motion,
  before and after the collision, is along one
  straight line.
  
• Glancing collisions involve an angle.
• A vector diagram can be used to represent the
  momentum for a glancing collision.

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Vector Diagrams

• Use the three vectors and construct a triangle.

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Vector Diagrams

• Use the appropriate expression to determine the
  unknown variable.

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Vector Diagrams

• Total vector momentum is conserved. You could
  break each momentum vector into an x and y
  component.
  
• px before px after
• py before py after
• You would use the x and y components to determine
  the resultant momentum for the object in
  question
  
• Resultant momentum

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Vector Diagrams

• Right triangle trigonometry can be used to solve
  this type of problem

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Vector Diagrams

• Pythagorean theorem
• If the angle ? for the direction in which the
  cars go in after the collision is known, you can
  use sin, cos, or tan to determine the unknown
  quantity. Example determine final velocity vT
  if the angle is 25.

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Vector Diagrams

• To determine the angle at which the cars go off
  together after the impact

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Special Condition

• When a moving ball strikes a stationary ball of
  equal mass in a glancing collision, the two balls
  move away from each other at right angles.
  
• ma mb
• va 0 m/s

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Special Condition

• Use the three vectors to construct a triangle.

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Special Condition

• Use the appropriate expression to determine the
  unknown variable.
  

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Rocket Propulsion (Jet Propulsion)

• As fuel burns and exhaust gases leave the rocket
  or jet engine, momentum is carried with them.
  
• To conserve momentum, the rocket or jet must gain
  the same amount of momentum in the opposite
  direction.

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Rocket Propulsion (Jet Propulsion)

• Thrust the magnitude (size) of the force
  exerted by an engine or rocket.
  
• Mass flow rate for fuel
• The direction of the thrust is opposite to the
  direction of the exhaust gases coming from the
  rocket or jet engine.

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Ballistic Pendulum

• In the ballistic pendulum lab, a ball of known
  mass is shot into a pendulum arm. The arm swings
  upward and stops when its kinetic energy is
  exhausted.
• From the measurement of the height of the swing,
  one can determine the initial speed of the ball.
  
  
• This is an inelastic collision. As always, linear
  momentum is conserved.

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Ballistic Pendulum
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Ballistic Pendulum

• Potential energy of ball in gun

• Ball embeds in pendulum

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Ballistic Pendulum

• Pendulum rises to a maximum height

• Solving for the initial speed of the projectile
  we get

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Helpful Websites

• Physics Classroom Momentum and Collisions
• ExploreScience Two Dimensional Collisions

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Elastic Collision Example

• Example mass 1 and mass 2 collide and bounce
  off of each other
  
• Momentum equation
• Kinetic energy equation
• v1 and v2 velocities before collision
• v1? and v2? velocities after collision
• Velocities are or to indicate directions.

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Elastic Collision Example

• Working with kinetic energy
• 0.5 cancels out.

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Elastic Collision Example

• The velocity terms are perfect squares and can be
  factored
  
• a2-b2 (a b)(a b)
• We will use this equation later.

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Elastic Collision Example

• Momentum equation

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Elastic Collision Example

• Both the kinetic energy and momentum equations
  have been solved for the ratio of m1/m2.
  
• Set m1/m2 for kinetic energy equal to m1/m2 for
  momentum

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Elastic Collision Example

• Get all the v1 terms together and all the v2
  terms together
  
• Cancel the like terms

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Elastic Collision Example

• Rearrange to get the initial and final velocities
  back together on the same side of the equation
  
• This equation can be solved for one of the two
  unknowns, then substituted back into the
  conservation of momentum equation.