Using the

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DEFINITION OF AVERAGE POWER Average power is the
rate at which work is done, and it is obtained by
dividing the work by the time required to
perform the work.
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Power of a human being
An interesting calculation is the average power
output of a human being. This can be determined
from the amount of energy we consume in a day in
the way of food. Most of us take in something
like 2500 "calories" in a day, although what we
call calories is really a kilocalorie. Use this
as our energy output per day Why is this a valid
assumption?. Take the 2.5 x 106 cal and
convert to joules, using the conversion factor
1 cal 4.186 J This gives roughly 1 x 107 J.
Figuring out our average power output, we simply
divide the energy by the number of seconds in a
day, 86400, which gives a bit more than 100 W.
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6.7 Power
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A ballistic pendulum

• A ballistic pendulum is a device used to measure
  the speed of a bullet. A bullet of mass m is
  fired at a block of wood (mass M) hanging from a
  string. The bullet embeds itself in the block,
  and causes the combined block plus bullet system
  to swing up a height h. What is v0, the speed of
  the bullet before it hits the block?
• Simulation of the ballistic pendulum
• We will work backwards to find an expression for
  v0.

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Mechanical energy conservation?

• Define the zero level for gravitational potential
  energy so both the bullet and the block have zero
  initial gravitational potential energy. Is it
  then correct to set the bullets kinetic energy
  before the collision to the bullet blocks
  gravitational potential energy when the bullet
  block reach their highest point after the
  collision?
• Yes
• No

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Mechanical energy conservation?

• Mechanical energy is not conserved in the
  collision.
• A completely inelastic collision occurs, in which
  some (in fact, most!) of the mechanical energy is
  transformed into thermal energy when the bullet
  embeds itself in the block.
• Is anything conserved during the collision?

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Mechanical energy conservation?

• Mechanical energy is not conserved in the
  collision.
• A completely inelastic collision occurs, in which
  some (in fact, most!) of the mechanical energy is
  transformed into thermal energy when the bullet
  embeds itself in the block.
• Is anything conserved during the collision?
• Yes, momentum is conserved over the small time
  period during which the collision occurs.

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Mechanical energy conservation!

• Mechanical energy is conserved in the pendulum
  motion after the collision.
• Work backwards, starting with the swing of the
  pendulum just after the collision until it
  reaches its maximum height, h.
• Write out the five-term energy-conservation
  equation.
• Eliminate terms that are zero.
• Substitute expressions for these terms

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Mechanical energy conservation!

• Solve for the speed of the pendulum at its lowest
  point. We get the familiar expression
• This is the final velocity of the system after
  the collision.
• In our collision analysis, then, we have

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Analyzing the collision

• Apply momentum conservation to the collision.
• Momentum beforehand momentum afterwards
• Bring in our previous result
• Solve for the speed of the bullet before the
  collision

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A numerical example

• If we use the following
• Mass of the bullet m 30 grams
• Mass of the block M 870 grams
• Maximum height h 0.74 m
• We find that

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How much mechanical energy is lost?

• Look at the ratio of the kinetic energy after the
  collision to the kinetic energy before the
  collision
• From momentum conservation
• In our numerical example, this ratio is 0.033.
  3.3 of the mechanical energy remains. 96.7 is
  lost!

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Worksheet, page 2

• Two balls hang from strings of the same length.
  Ball A, with a mass of 4 kg, is swung back to a
  point 0.8 m above its equilibrium position. Ball
  A is released from rest and swings down and hits
  ball B. After the collision ball A rebounds to a
  height of 0.2 m above its equilibrium position,
  and ball B swings up to a height of 0.05 m.

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Speed of ball A, before

• How fast is ball A going, just before the
  collision?
• Apply energy conservation.
• Eliminate three of the terms.

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Speed of the balls, after the collision

• We can use the same equation afterwards.
• For ball A afterwards
• For ball B afterwards

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What is the mass of ball B?

• Find the mass of ball B.
• 4 kg
• 8 kg
• 12 kg
• 16 kg
• 24 kg
• None of the above

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Find the mass of ball B

• Apply momentum conservation.
• How do we account for the fact that momentum is a
  vector?

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Find the mass of ball B

• Apply momentum conservation.
• How do we account for the fact that momentum is a
  vector?
• Choose a positive direction (to the right), so
  the velocity of ball A after the collision is
  negative.

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What kind of collision?

• Relative speed before the collision 4 m/s
• Relative speed after the collision 3 m/s
• Elasticity .
• This is less than 1, so the collision is
  inelastic.

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What kind of collision?

• Kinetic energy before the collision 32 J
• Kinetic energy after the collision 8 J 12 J
  20 J
• The kinetic energy is smaller after the
  collision, so the collision is inelastic. It is
  not completely inelastic, because the two balls
  do not stick together after the collision.

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Collisions in two dimensions

• The Law of Conservation of Momentum applies in
  two and three dimensions, too. To apply it in
  2-D, split the momentum into x and y components
  and keep them separate. Write out two
  conservation of momentum equations, one for the x
  direction and one for the y direction.
• For example
• Simulation of a 2-D collision

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Baseball and basketball

• Dropping a basketball with a baseball balanced on
  top can produce interesting results. Watch the
  demonstration.
• The basketball has three times the mass of the
  baseball.
• How do we explain what we see? Lets start by
  analyzing this as an elastic collision.
• Simulation of the baseball/basketball situation

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Baseball and basketball

• The two balls drop through the same height h, so
  they are both traveling down with a speed v when
  the basketball hits the ground.
• Assumption the basketballs collision with the
  floor is elastic, so after colliding with the
  floor the basketballs velocity is v directed up.

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Baseball and basketball

• Now we have a collision between the baseball, of
  mass m and velocity v down, and the basketball,
  of mass 3m and velocity v up.
• Observation What does the basketball do after
  the collision?

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Baseball and basketball

• Now we have a collision between the baseball, of
  mass m and velocity v down, and the basketball,
  of mass 3m and velocity v up.
• Observation What does the basketball do after
  the collision?
• The basketball is essentially at rest after the
  collision.
• Apply momentum conservation to find the speed of
  the baseball. Choose up to be positive.
• vf ?

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Baseball and basketball

• Now we have a collision between the baseball, of
  mass m and velocity v down, and the basketball,
  of mass 3m and velocity v up.
• Observation What does the basketball do after
  the collision?
• The basketball is essentially at rest after the
  collision.
• Apply momentum conservation to find the speed of
  the baseball. Choose up to be positive.
• vf 2v (this solution also conserves
  kinetic energy)

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Baseball and basketball

• How high does the baseball go?
• When it was dropped from a height h, the baseball
  acquired a speed v
• If the baseball is fired up with a speed 2v, how
  high does it go?

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Baseball and basketball

• How high does the baseball go?
• When it was dropped from a height h, the baseball
  acquired a speed v
• If the baseball is fired up with a speed 2v, how
  high does it go?
• H 4h. The baseball goes four times as high. It
  would go even higher, to 9h, if its mass was
  negligible compared to the basketballs mass.

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Which collision?

• You are driving at high speed along a divided,
  multi-lane highway when you see your evil twin,
  driving an identical car, going the wrong way and
  coming directly toward you. You both slam on your
  brakes, but it's too late to stop and there is
  about to be a collision. At the last instant you
  spot a large, solid immovable object by the side
  of the road.
• Assume the speed when you collide is the
  same whether you hit your evil twin or the
  immovable object, and that your evil twin is
  going at the same speed you are. Either collision
  is a head-on collision.
• It is better for you (in terms of you surviving
  the crash) to
• 1. Hit your evil twin
• 2. Hit the immovable object
• 3. Neither one - they're equivalent as far as
  you're concerned

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Whiteboard