Science 111

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Science 111

• Chapter 3
• Momentum and Energy

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Why do cannons have long barrels?

• Two cannons, same gunpowder, same diameter, same
  cannonball, only different in the length of their
  barrels
• The shell from the cannon with the longer barrel
  will come out traveling faster.
• Why?

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Why do cannons have long barrels?

• The same force acts in either case and will cause
  the same acceleration up to a point.
• Once the cannonball exits the barrel, the hot,
  exploding gases escape into the air and are no
  longer pushing on the shell.
• For the longer barrel, we can say either the
  force acted over more time or for a greater
  distance.

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Impulse and Momentum

• The time during which a force acts multiplied by
  the amount of the force gives a quantity we will
  call impulse.
• The impulse measures the amount of momentum
  given to the shell.
• The cannon with the longer barrel exerted a
  greater impulse and the shell ends up with more
  momentum (and hence speed).

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Work and Energy

• The distance over which a force acts multiplied
  by the amount of force gives a quantity we will
  call work.
• The work measures the amount of energy given to
  the shell.
• The cannon with the longer barrel did a greater
  amount of work and the shell ends up with more
  energy (and hence speed).

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Three methods?

• So, let me get this straight, now we have three
  ways of solving problems?
• Acceleration, velocity, position calculations
• Impulse and momentum calculations
• Work and energy calculations
• Uh, yeah, sorry.
• But that which doesnt kill you

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Momentum

• The momentum of an object equals
• (mass of object) x (velocity of object)
• mv
• Momentum measures how strongly an object is
  moving in a direction.
• Momentum is a vector quantity.

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Impulse

• The impulse delivered by a force equals
• (amount of force) x (time duration of force)
• Ft
• It is a measure of the total effect a force has
  had in a particular direction.
• Impulse, too, is a vector quantity.

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Section 3.3Impulse-Momentum Relationship

• The impulse delivered by a force will equal the
  change in momentum of the object acted on by the
  force.
• Written mathematically as
• Ft ?(mv)
• ? Greek letter delta, used as shorthand to
  mean the change in.

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Impulse-Momentum Relationship

• Ft ?(mv)
• Results obtained using this formula are the same
  as what you would get by, say, using F to
  calculate acceleration a, then a and t to figure
  out how v changes.
• That is, this is equivalent to Newtons 2nd Law
  in fact this is close to how Newton originally
  wrote the second law.

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Ft m?v

• The change in the momentum on an object is
  usually due to a change in velocity, not a change
  in mass, so ?(mv) m ?v.
• When the momentum of an object changes, it can
  always be explained by some force F acting for
  the necessary time t.

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• Stopping a moving truck requires a large change
  in momentum, a large impulse.
• mV represents a large change in velocity.
• The haystack - which exerts a small force - will
  have to act over a long time (fT).

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• Stopping the truck with a sturdy wall will
  require the same impulse from the wall as the
  haystack.
• The wall exerts a very large force over a very
  short time (Ft).

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The cannon

• Recall the long-barreled cannon question.
• Extending the barrel increases the time that the
  force acts on the shell.
• A greater impulse is achieved.
• So the shell or cannonball gets a greater
  increase in its momentum.
• Greater mv means greater v.

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Catching an Egg

• How do you catch a thrown egg without it
  breaking?
• To catch it means bringing it to a stop,
  decreasing its momentum to zero.
• Some amount of impulse will be required depending
  on the eggs mass and approach speed, you cant
  change that.

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Catching an Egg (2)

• But that impulse can be delivered as a large
  force for a short time, or a small force for a
  long time.
• It will break if the force is too large.
• So, you want a small force for a long time.
• Demonstration???

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Sec. 3.4 - Conservation of Momentum

• Two objects interact via a force.
• Maybe a bat and a ball.
• They exert equal but opposite forces on each
  other (Newtons Third Law).
• Forces will be for the same length of time.
• Bat and ball will receive equal impulses (but in
  opposite directions).

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Equal momentum changes

• The ball will gain momentum in one direction
  while the bat will gain momentum in the opposite
  direction.
• The total (vector total) change in momentum is
  zero.
• The total (vector sum) of momentum does not
  change.

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Conservation of Momentum

• Physicists say that momentum is conserved.
• The total amount is always the same.
• It always works that way, no exceptions.
• Again, momentum conservation is a direct
  consequence of Newtons third law.

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Warnings

• Momentum conservation works only if
• You remember that momentum is a vector quantity.
• (Momentum (mv) leftwards plus (mv) rightwards is
  equal to zero total momentum.)
• You must include everything that is changing
  momentum.
• (Like bat, ball, and batter.)

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Another Warning

• The same change in momentum is not the same
  change in speed.
• When the bat hits the ball, for instance, the
  ball will likely gain more speed than the bat
  loses.
• In equal momentum changes, the object with the
  least mass will have the greatest change in speed.

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Example Balloon

• Balloon squirting air and moving (like in figure
  2.24 in the text).
• The balloon gains forward momentum, the air
  molecules gain backwards momentum.
• The total (vector) change in momentum is zero.

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Example Rocket

• A rocket lifting off the ground.
• Explanation the same as for the balloon (they are
  both rockets!).
• Rockets gain of forward momentum equals the
  total gain of reverse momentum by the exhaust.

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Example Cannon

• Before the cannon fires, the total momentum is
  zero (nothing is moving)
• After the cannon fires, the cannon and cannonball
  both have momentum.

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Cannon (II)

• Huh? No momentum before, lots after, that doesnt
  sound like momentum conservation!
• But it is!
• The cannonball has rightward momentum, maybe (20
  kg)(50 m/s) 1000 kg m/s.
• The cannon has leftward momentum, maybe (500
  kg)(2 m/s) 1000 kg m/s.

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Cannon (III)

• The vector total was zero both before and after
  the cannon shot.
• Individual momenta can change, but every change
  in one direction is compensated by an equal
  change of something elses momentum in the
  opposite direction.

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Cannon (revisited)

• Cannon and cannonball?
• Are we missing anything?
• Does anything else change momentum here?

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That Darn Cannon

• There were two omissions.
• The exhaust gases going forward as well as the
  cannonball.
• The cannon appeared dug into the ground a bit, so
  the whole Earth (!) may have gained momentum
  backwards.

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Example Dropped Ball

• A ball is released and falls to the Earth.
• It had zero momentum when first released, but
  gained downward momentum as it fell.
• Where is my momentum conservation now?
• Well, urrr, let me think

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Dropped Ball (II)

• Why did the ball fall?
• Because the Earth pulled it downwards with
  gravity.
• So, by Newtons third law, the ball was pulling
  upward on the Earth.
• The Earth gains upward momentum.

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Dropped Ball (III)

• But the Earth just sits there, it cant gain as
  much momentum as the ball did.
• Yes, it can.
• Because of its huge mass, the Earth has to gain
  just a minuscule velocity upward to gain as much
  momentum as the ball did.

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Solving Problems

• We can solve problems using Fma, velocities,
  positions, etc.
• We can also use impulse and momentum.
• But the real breakthrough is momentum
  conservation.

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Momentum Conservation

• Momentum conservation allows us to solve problems
  just by comparing the initial and final states.
• We dont have to know what forces acted, or for
  how long, or in what directions.
• The total momentum before and after must be the
  same regardless.

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Example Car Crashes
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Head-On Collision

• If two cars have the same mass and approach at
  the same speed in a head-on collision, then the
  total momentum before the collision is zero.
• Assuming no significant outside forces act (which
  will probably be true), the total momentum after
  the collision is zero.

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Crash

• If the cars crumple and merge into one big lump,
  the lump will be stationary.
• If the cars rebound a bit, they will be moving
  away with equal but opposite velocities.
• Guaranteed, the total momentum must always be
  zero.

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Uneven Crash

• If one car is heavier than the other and they
  approach with equal speeds, the total momentum is
  not zero.
• After the collision, the total momentum will be
  the same non-zero value.

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Uneven Crumple

• If the total momentum was non-zero before the
  crash and the cars crumple into one big lump,
  that lump will be sliding with the speed and
  direction that give it the same momentum.
• It may then slow due to friction but the Earth
  and lump will have the same total amount of
  conserved momentum.

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Work and Energy

• Energy
• Matter is substance and energy is the mover of
  substance.
• Anything with the potential to make things move
  is said to contain energy.
• The transfer of energy into some form of motion
  is called work.

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Work

• When something is made to move, we say that work
  has been done.
• Forces acting on moving objects do work.
• If there is no motion, no work is done.
• Our technical definition of work may disagree
  with your everyday meaning of the word.

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Calculating Work

• The exact amount of work done by a force acting
  on a moving object is calculated by
• Work (force) x (distance moved) Fd
• This work represents how much energy has been
  transferred by the body exerting the force to the
  moving body.
• Unit of work or energy J joule Nm

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Positive and Negative Work

• If the force is pushing the object in the same
  direction it is moving, positive work is done.
• The object gains energy and the pusher loses
  energy.
• If the force is opposite the direction of motion,
  negative work is done.
• Object loses energy, pusher gains energy.
• Same direction W Fd
• Opposite directions W -Fd

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Sec. 3.6 Power

• The rate at which work is done, or the rate at
  which energy is transferred, is the power.
• A rate is an amount divided by a time.
• Power work/time energy/time
• Unit of power W watt J/s

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What?

• Many devices are rated in terms of their power,
  their wattage.
• A 100-watt light bulb converts electrical energy
  into light (and heat) at a rate of 100 joules per
  second.
• Work is being done, electrical forces exerted by
  the power company push on electrons making them
  go faster and they hit atoms to make light.

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Types of Energy

• An object contains energy if it can exert forces,
  do work, and make things move.
• There are a variety of common forms this energy
  takes within bodies.
• Recognizing these forms is crucial to
  understanding how things work.

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Kinetic Energy

• Kinetic energy is energy of motion.
• Any moving object contains energy because it can
  push against other objects - exerting forces
  while making them move.
• This is also the form of energy normally gained
  when an object has work done on it.

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Calculating Kinetic Energy

• The formula for calculating kinetic energy is KE
  (1/2) m v2.
• An object of mass (10 kg) moving at a speed of (4
  m/s) will have kinetic energy
• KE .5 (10) (42) 80 kg m2/s2 80 J
• All moving objects have kinetic energy, the more
  mass and faster, the more KE.

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Kinetic Energy vs Momentum

• KE is 1/2 mv2 momentum is mv
• Moving objects have both.
• But they are not the same.
• First, kinetic energy is a scalar, just a number
  representing the energy of motion.
• Momentum is a vector, a magnitude and a direction.

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KE vs Mtm (II)

• Momentum is transferred by forces, the momentum
  transfer is the force multiplied by the time
  (impulse).
• Kinetic energy is transferred by forces, the
  energy transfer is the force multiplied by the
  distance moved (work).
• Both are useful quantities.

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Potential Energy

• Another form of energy is called potential
  energy.
• It is energy possessed because of the position or
  configuration of something.
• Work must be done to give an object potential
  energy, then the object can do work and transfer
  that energy onwards.

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Gravitational Potential Energy

• Objects at a greater height have more potential
  energy.
• When they fall back down, their potential energy
  becomes kinetic energy which may become other
  forms depending on what they later run into.
• Gravitational PE formula PE mgh
• Mass x gravity (9.8 m/s2) x height up

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Electrical Potential Energy

• Electrically charged particles can have potential
  energy associated with electrical forces.
• The amount of electrical potential energy depends
  on the voltage.
• Well learn more about this in chapter 9.

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Thermal Energy

• Suppose an object is not moving.
• So it has no kinetic energy.
• But if you could look at the atoms making up the
  object using a powerful microscope, youd see
  that the atoms are moving.
• The atoms have kinetic energy.

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Thermal Energy (II)

• When the kinetic energy is stored in the random
  motions of microscopic components, we dont call
  it kinetic energy.
• Instead we call it thermal energy.
• It is still energy that can be used to do work,
  but not as accessible as normal KE.
• Temperature is related to the amount of thermal
  energy, details in chapter 7.

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Nuclear Energy

• The nuclei of atoms (the protons and neutrons)
  can exert potent forces and hence contain
  substantial energy.
• We wont get into details here (but you may in
  Science 112).
• The famous formula E mc2 is related to the
  stored nuclear energy.

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Still other forms of energy

• Wave Energy
• Includes the energy in sound waves and light.
• Elastic Potential Energy
• Energy stored in the deformation of an object,
  like a compressed spring.
• Chemical Energy
• Energy available if chemicals are allowed to
  combine (like gasoline and oxygen).

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Work-Energy Theorem

• The work done on an object equals the change in
  its kinetic energy.
• Equation Fd ?(KE)
• Fd is the work done, it changes the kinetic
  energy of the object acted on (increasing it if
  positive work, decreasing it if negative).
• The KE may later become other forms of energy.

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Mtm and KE (again)

• Force x Time (impulse) change in momentum
• Force x Distance (work) change in kinetic
  energy
• Depending on whether you know the time or the
  distance for which the force acted may determine
  whether you analyze a problem using momentum or
  energy.

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What did the work on the cannonball?

• What actually exerted the force on the
  cannonball?
• What was touching it?
• The hot, expanding gases did the work to push the
  cannonball forward.

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What does work on the ball?

• Chemical forces do work on the arm muscle.
• The muscle does work on the arm.
• The arm does work on the hand.
• The hand does the work pushing the ball.

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What does work on the ball (while it is in the
air)?

• Gravity does work, changing the velocity and
  kinetic energy during the flight.
• Air-resistance will also do some (negative) work
  on the ball.

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What does work to slow the truck
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But the wall doesnt move.

• The wall exerts the force and hence does the work
  to stop the truck.
• Wait! The wall cant do work because the force is
  exerted on something that isnt moving (the front
  of the truck).
• So, isnt work Fd 0?
• Then what does stop the truck?

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No immovable objects

• When things collide, there is always some amount
  of give.
• Everything will move or deform, at least a
  little, when a new force acts on it.
• The collision force is enormous, the movement
  distance is small, but the product Fd will be
  equal to the change in kinetic energy of the
  truck.

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What does the work to hold the lamp atop the
table?

• Easy, the table.
• No!
• No work is done here.
• No motion so no work.

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Lamp on a Table

• The table does exert a force on the lamp.
• There may have been some work when the lamp was
  first placed there and the table deformed a
  little.
• But no work, no transfer of energy, is occurring
  while the lamp just sits there.

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It feels like work to me

• But if you hold a lamp in your outstretched arm,
  it certainly feels like you do work.
• Yes, but the work is not at your hand, its in
  your shoulder and muscles.
• There are vibrations and chemical reactions
  needed to maintain your arm position
  out-stretched - but there is no energy exchange
  (work) between your hand and lamp!

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Sec. 3.9 Conservation of Energy

• Whenever one object gains energy (in any form),
  another object will lose exactly as much energy.
• The total amount of energy is always the same.
• Energy cannot be created or destroyed.

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Conservation of Energy

• Energy can change forms.
• Energy can move from one body to another.
• But the total amount of energy never changes.
• Total energy (not kinetic energy) is conserved.

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Solving Problems

• Knowing that the total energy is conserved often
  allows one to quickly solve problems.
• We will mostly use conservation of energy to
  figure things out rather than the work-energy
  theorem.
• Lets do some examples

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The falling object converts its gravitational
potential energy into kinetic energy. Gravitation
al PE represents the work that will be done by
gravity as the object moves up or
down. Calculations like this mgh (1/2)mv2
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Falling, PE gt KE

• When the object hits the stake, more forces are
  exerted (along with some movement) so there is a
  further energy exchange.
• The kinetic energy becomes mostly thermal energy
  (object and stake get hotter), some energy may
  end up in other forms like sound.

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What are the forms of energy?
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Inside Sun Nuclear Energy Thermal
Energy Sunlight Light or Wave
Energy Plants Chemical Energy
Fossil Fuels, Animals, Food Chemical
Energy Power Electrical Energy
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A ball is thrown off a cliff

• If the ball is thrown at 50 mph, what direction
  should it be thrown so that it will have the
  greatest speed when hitting the ground below?
• Throw it straight downwards
• Throw it straight upwards
• Throw it horizontally
• Will be the same in all cases

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Answer (d)

• Ignoring air-resistance, conservation of energy
  is the key to answering this.
• The ball, when thrown, will have both the same
  kinetic energy and potential energy no matter
  which direction its thrown.
• Hence all will have the same kinetic energy (and
  speed) when hitting the ground.

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This time with drag

• Now lets include air-resistance.
• Air resistance always does negative work because
  it acts opposite to the direction the object is
  moving through the air.
• The longer the path, the more energy that will be
  lost to the air (air-resistance causes the air to
  get warmer, thermal energy).
• So path (a), the shortest, will have the most
  energy remaining and the greatest impact speed.

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Football Sleds

• Three football players in practice all do the
  same exercise.
• The run full speed and hurtle their bodies
  against the same heavy sled.
• The impact causes the sled to slide a ways.
• Given the following information, which player
  will probably cause the sled to move the furthest?

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Player Stats

• Player 1
• Mass 100 kg Speed 7 m/s
• Player 2
• Mass 120 kg Speed 5 m/s
• Player 3
• Mass 80 kg Speed 8 m/s

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Idea Compare Energies

• Whoever can give the most energy to the sled will
  cause it to move the furthest, right?
• P1 KE (.5)(100 kg)(7 m/s)2 2450 J
• P2 KE (.5)(120 kg)(5 m/s)2 1500 J
• P3 KE (.5)(80 kg)(8 m/s)2 2560 J
• Player 3 has the most energy, and should cause
  the sled to go furthest.

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Wrong? I hate physics!

• That analysis would be correct if all the
  players kinetic energy could be transferred to
  the sled as kinetic energy.
• There will be conservation of energy but the
  players original KE will end up in a variety of
  forms, most notably thermal energy and
  deformation (elastic) potential energies.

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Momentum

• The player and sled just before the collision
• and just after the collision will have the same
• total momentum (since we are considering
• only a short time interval, we can be confident
• that there are no external impulses messing
• up our momentum bookkeeping).

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Momentum Calculations

• Assuming the sled has a mass of 150 kg and the
  player and sled move together right after the
  collision,
• (M)(V) (150 kg)(0) (M150 kg)Vfinal
• This gives Vfinal (MV)/(M150)

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Finishing the Calculation

• Player 1
• V1 (100)(7 m/s) / (250) 2.8 m/s
• Player 2
• V2 (120)(5 m/s) / (270) 2.2 m/s
• Player 3
• V3 (80)(7 m/s) / (230) 2.4 m/s
• Player 1 is going to push it furthest.

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End of Chapter 3

• It takes a lot of practice to know when energy or
  momentum conservation can be applied.
• Skip sections 3.10 and 3.11.
• End of Chapter 3