The Physics of Renewable Energy

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Collisions
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Objectives

• Describe a perfectly inelastic collision.
• Apply conservation of momentum to collisions in
  one dimension.
• Define and describe an elastic collision.
• Describe the possible outcomes that result from
  the collision of one moving ball with one
  stationary ball when their masses are equal or
  unequal.
• Determine if a collision is elastic or not.

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Physics terms

• collision
• inelastic collision
• perfectly inelastic collision
• elastic collision

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Equations
Conservation of momentum
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Equations
Conservation of kinetic energy (elastic
collisions only!)
Conservation of momentum
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An interaction between two or more bodies in
motion is a collision.
What is a collision?
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What is a collision?
Pools balls bouncing off of each other is one
example.
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What is a collision?
Atoms repelling each other in a gas is a
more important example.
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Even a comet rounding the Sun is a collision in
the sense of fundamental conservation laws.
What is a collision?
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What is conserved?
What is conserved in a collision, and why? To
answer this question, lets look at what happens
during a collision.
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What happens in a collision?
What is happening in this collision between two
balls? What might happen next? There are a
couple possibilities.
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What cant happen in a collision?
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How do the velocities change?
Here is one possible result of the collision.
Examine the velocities shown in the figure. How
do the velocities change?
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What forces are generated?
What is true about these forces? Which of
Newtons laws applies here?
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What forces are generated?
What is the NET force on this system?
system boundary
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What forces are generated?
This is a closed system. No external forces act
on the system, and the internal forces cancel
each other. So what is conserved?
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Conservation laws
Momentum Energy
The momentum before the collision equals the
momentum after the collision. The energy before
the collision equals the energy after the
collision. Butthe energy may be transformed.
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Three types of collisions

• Perfectly inelastic collision The objects stick
  together.
• Inelastic collision These collisions are
  somewhat bouncy.
• Elastic collisions These collisions are
  perfectly bouncy.

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Three types of collisions
Momentum is conserved in all three types of
collisions.
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Conservation of momentum
In a closed system, the momentum BEFORE the
collision equals the momentum AFTER the collision
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Conservation of momentum
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Conservation of momentum
A green ball mass m1 collides with a red ball of
mass m2
m1 m2
pi pf m1 vi1 m2 vi2 m1
vf1 m2 vf2
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Conservation of momentum
vi1
In this collision the red ball is initially at
rest.
m1 m2
pi pf m1 vi1 m2 vi2 m1
vf1 m2 vf2
Which term can we get rid of in this equation?
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Conservation of momentum
vi1
The second term equals zero. We can get rid of it.
m1 m2
pi pf m1 vi1 m2 vi2 m1
vf1 m2 vf2 m1 vi1 m1 vf1
m2 vf2
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Conservation of momentum
The balls stick together after this collision.
How do we show that in this equation?
vi1
m1 m2
pi pf m1 vi1 m2 vi2 m1
vf1 m2 vf2 m1 vi1 m1 vf1
m2 vf2
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Conservation of momentum
vi1
Now there is only one final velocity.
m1 m2
pi pf m1 vi1 m2 vi2 m1
vf1 m2 vf2 m1 vi1 m1 vf1
m2 vf2
m1 vi1 (m1 m2) vf
vf
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Solving the problem
A green ball has a mass of 5.0 kg and a velocity
of 10 m/s. A red ball has a mass of 15 kg and
is initially at rest. The balls collide and
stick together. What is their resulting
velocity?
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Solving the problem
A green ball has a mass of 5.0 kg and a velocity
of 10 m/s. A red ball has a mass of 15 kg and
is initially at rest. The balls collide and
stick together. What is their resulting
velocity? Before you plug in numbers, can you
make any predictions about this velocity?
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Solving the problem
A green ball has a mass of 5.0 kg and a velocity
of 10 m/s. A red ball has a mass of 15 kg and
is initially at rest. The balls collide and
stick together. What is their resulting
velocity? Does this answer make sense?
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Assessment

• Define a perfectly inelastic collision.

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Assessment

• Define a perfectly inelastic collision.

• In a perfectly (or totally) inelastic collision,
  the two objects stick together after the
  collision. There is only one final, shared
  velocity.

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Assessment

• A 10 kg puck moving with a velocity of 3.0 m/s
  strikes a second stationary 10 kg puck. The
  pucks stick together after the collision.

• What is their combined velocity after the
  collision?

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Assessment

• A 10 kg puck moving with a velocity of 3.0 m/s
  strikes a second stationary 10 kg puck. The
  pucks stick together after the collision.

• What is their combined velocity after the
  collision?

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Assessment

• A 10 kg puck moving with a velocity of 3.0 m/s
  strikes a second stationary 10 kg puck. The
  pucks stick together after the collision.

• What is their combined velocity after the
  collision?

The moving mass doubles, so the velocity is half
as big.
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Assessment

• A 10 kg puck moving with a velocity of 3.0 m/s
  strikes a second stationary 10 kg puck. The
  pucks stick together after the collision.

• What was the kinetic energy of the system before
  the collision?
• After the collision?

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Assessment

• A 10 kg puck moving with a velocity of 3.0 m/s
  strikes a second stationary 10 kg puck. The
  pucks stick together after the collision.

• What was the kinetic energy of the system before
  the collision?
• After the collision?

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Assessment

• A 10 kg puck moving with a velocity of 3.0 m/s
  strikes a second stationary 10 kg puck. The
  pucks stick together after the collision.

• What was the kinetic energy of the system before
  the collision?
• After the collision?

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Assessment

• A 10 kg puck moving with a velocity of 3.0 m/s
  strikes a second stationary 10 kg puck. The
  pucks stick together after the collision.

• What was the kinetic energy of the system before
  the collision?
• After the collision?

Was kinetic energy conserved?
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Elastic collisions
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Review conservation laws
closed system
Momentum
The momentum before the collision equals the
momentum after the collision. The energy before
the collision equals the energy after the
collision.
Energy
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Review types of collisions

• Perfectly inelastic collisions The objects
  stick together.
• Inelastic collisions These collisions are
  somewhat bouncy.
• Elastic collisions These collisions are
  perfectly bouncy.

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Review types of collisions
Momentum is conserved in all three collisions.
What is different about them?
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Elastic vs. inelastic collisions
Momentum is conserved in all three collisions.
What is different about them?
One of them ALSO conserves kinetic energy. Which
one?
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Elastic vs. inelastic collisions

• Elastic collisions the total kinetic energy IS
  conserved.
• Inelastic collisions kinetic energy is NOT
  conserved. Some of it is transformed to other
  types of energysuch as thermal energy or sound
  energy.

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Inelastic collisionsconserve momentumbut NOT
kinetic energy.
Inelastic collisions
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Inelastic collisions
All real collisions between macroscopic
objects are inelastic.
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Elastic collisions
If these collisions were totally elastic, the
ball would bounce forever.
So if a bouncing ball isnt really elastic, then
what is?
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Examples of elastic collisions
Collisions between gas molecules are elastic.
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Examples of elastic collisions
Near-miss collisions between the comet and the
Sun can also be treated as elastic.
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Examples of elastic collisions
Collisions between billiard balls are nearly
elastic.
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Examples of elastic collisions
Collisions between steel balls in this popular
toy are nearly elastic.
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An elastic collisionconserves momentumand
kinetic energy.
Elastic collisions
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Conservation of kinetic energy
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Case 1 m1 m2
m2
m1
What happens when masses are equal?
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Case 1 results m1 m2
v2 v0
v1 0
m1
m2
The green ball stops. All the velocity of the
green ball is transferred to the red ball.
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Case 1 m1 m2
The equal-mass balls in this toy illustrate this
situation. The incident ball stops and the
target ball takes off with all the momentum and
kinetic energy.
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Case 2 m1lt m2
m1
m2
What happens when the green ball has LESS mass?
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Case 2 results m1lt m2
m1
m2
When the green ball has LESS mass than the red
ball, the green ball bounces backwards . . .
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Case 2 results m1lt m2
m1
m2
. . . and the red ball goes
forward. But the red balls velocity is less than
the green balls initial velocity.
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Case 2 results m1ltlt m2
m2
m1
The larger the ratio of the masses . . .
the less velocity is transferred to the red ball
and the greater the rebound speed of the green
ball.
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Case 3 m1gt m2
m1
m2
What happens when the green ball has MORE mass?
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Case 3 results m1gt m2
m1
m2
When the green ball has more mass, both balls
move forward. But the red ball moves faster, and
the green ball loses speed.
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Case 3 results m1gtgt m2
m1
m2
When the green ball has MUCH more mass, it is
barely affected by the collision. It moves
forward at ALMOST its original velocity.
What happens to the red ball?
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Case 3 results m1gtgt m2
m1
m2
The red ball takes off at high speedalmost twice
the initial velocity of the green ball!
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Summarizing the three cases
The balls swap their velocities. The less
massive incoming ball bounces backward. The more
massive incoming ball continues forward, and the
less massive target ball moves forward even
faster.

• m1 m2
• m1 lt m2
• m1 gt m2

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Elastic collisions
Kinetic energy is conserved.
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Elastic collisions
Kinetic energy is conserved. Momentum is also
conserved.
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Patterns in the velocities
What do you notice when you compare approach
velocity (v1 2 m/s) to the final velocities?
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Patterns in the velocities
The approach speed v1 always equals the
separation speed (vf2 vf2). Elastic
collisions are perfectly bouncy.
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Elastic collisions

In an elastic collision, the total kinetic energy
before the collision is the same as the total
kinetic energy after the collision.
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Real (inelastic) collisions
gt
In an inelastic collision, the total kinetic
energy after the collision is less because some
energy has been converted to other forms.
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Comparing inelastic and elastic collisions
gt
Momentum is always conserved in
collisionswhether they are elastic or inelastic!
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Comparing inelastic and elastic collisions
gt
Momentum is always conserved in
collisionswhether they are elastic or inelastic!
Kinetic energy is only conserved in elastic
collisions.
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Assessment

• Define the difference between elastic and
  inelastic collisions.

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Assessment

• Define the difference between elastic and
  inelastic collisions.

An elastic collision is one in which both
momentum and kinetic energy are conserved. An
elastic collision is bouncy. The objects never
stick together.
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Assessment

• A 10 kg ball moving with a velocity of 3.0 m/s
  strikes a stationary 10 kg ball. If the
  collision is elastic, what are the velocities of
  the two balls after the collision?
• Hint these balls have equal mass.

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Assessment

• A 10 kg ball moving with a velocity of 3.0 m/s
  strikes a stationary 10 kg ball. If the
  collision is elastic, what are the velocities of
  the two balls after the collision?

• The balls swap velocities. The first ball stops,
  and the second ball moves off with a speed of
  3.0 m/s after the collision.

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Assessment

• A 1.0 kg ball moving with a velocity of 1.0 m/s
  strikes a stationary 3.0 kg ball. After the
  collision, the two balls stick together and move
  at 0.25 m/s. Was the collision elastic?

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Assessment

• A 1.0 kg ball moving with a velocity of 1.0 m/s
  strikes a stationary 3.0 kg ball. After the
  collision, the two balls stick together and move
  at 0.25 m/s. Was the collision elastic?

If the collision is elastic, then Eki will equal
Ekf. Kinetic energy before the collision
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Assessment

• A 1.0 kg ball moving with a velocity of 1.0 m/s
  strikes a stationary 3.0 kg ball. After the
  collision, the two balls stick together and move
  at 0.25 m/s. Was the collision elastic?

If the collision is elastic, then Eki will equal
Ekf. Kinetic energy before the collision Kinet
ic energy after the collision
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Assessment

• A 1.0 kg ball moving with a velocity of 1.0 m/s
  strikes a stationary 3.0 kg ball. After the
  collision, the two balls stick together and move
  at 0.25 m/s. Was the collision elastic?

If the collision is elastic, then Eki will equal
Ekf. Kinetic energy before the collision Kinet
ic energy after the collision
The collision is not elastic. Some Ek was
losttransformed into other kinds of energy.
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Assessment

• A 1.0 kg ball moving with a velocity of 1.0 m/s
  strikes a stationary 3.0 kg ball. After the
  collision, the two balls stick together and move
  at 0.25 m/s. Was the collision elastic?

There is a simpler way to tell that this
particular collision is NOT elastic. Elastic
collisions are perfectly bouncy. These balls did
not bounce at all.