Physics 151: Lecture 18 Today

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Physics 151 Lecture 18Todays Agenda

• Topics
• Review of momentum conservation
• 2-D Collisions
• Systems of particles

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Lecture 17 ACT 4

• The law of conservation of momentum applies to a
  collision between two bodies if
• they exert forces on each other respectively
  proportional to their masses.
• they exert forces on each other respectively
  proportional to their velocities.
• their accelerations are proportional to their
  masses.
• they exert equal and opposite forces on each
  other.

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Lecture 18Review problem qualitative

• Two boys in a canoe toss a baseball back and
  forth. What effect will this have on the canoe?
  Neglect (velocity-dependent) frictional forces
  with water or air.
• a. None, because the ball remains in the canoe.
• b. The canoe will drift in the direction of the
  boy who throws the ball harder each time.
• c. The canoe will drift in the direction of the
  boy who throws the ball with less force each
  time.
• d. The canoe will oscillate back and forth always
  moving opposite to the ball.
• e. The canoe will oscillate in the direction of
  the ball because the canoe and ball exert forces
  in opposite directions upon the person throwing
  the ball.

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Lecture 17, ACT 3Momentum Conservation

• Two balls of equal mass are thrown horizontally
  with the same initial velocity. They hit
  identical stationary boxes resting on a
  frictionless horizontal surface.
• The ball hitting box 1 bounces back, while the
  ball hitting box 2 gets stuck.
• Which box ends up moving fastest ?

(a) Box 1 (b) Box 2 (c)
same
2
1
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Lecture 17Review problem

• A 3.0-kg mass is sliding on a horizontal
  frictionless surface with a speed of 3.0 m/s when
  it collides with a 1.0-kg mass initially at rest
  as shown in the figure. The masses stick together
  and slide up a frictionless circular track of
  radius 0.40 m. To what maximum height, h, above
  the horizontal surface will the masses slide?

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Inelastic collision in 2-D

• Consider a collision in 2-D (cars crashing at a
  slippery intersection...no friction).

v1
m1 m2
m1
m2
v2
before
after
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Inelastic collision in 2-D...

• There are no net external forces acting.
• Use momentum conservation for both components.

v1
V (Vx,Vy)
m1 m2
m1
m2
v2
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Inelastic collision in 2-D...

• So we know all about the motion after the
  collision !

V (Vx,Vy)
Vy
?
Vx
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Elastic Collisions
See text 9.4

• Elastic means that energy is conserved as well as
  momentum.
• This gives us more constraints.
• We can solve more complicated problems !!
• Billiards (2-D collision).
• The colliding objectshave separate motionsafter
  the collision as well as before.
• Start with a simpler 1-D problem.

Before
After
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Elastic Collision in 1-D
See text 9.4
m2
m1
before
v1,b
v2,b
x
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Elastic Collision in 1-D
See text 9.4
m1
m2
before
v1,b
v2,b
x
m1v1,b m2v2,b m1v1,a m2v2,a
after
v2,a
v1,a
Suppose we know v1,b and v2,b We need to solve
for v1,a and v2,a
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Elastic Collision in 1-D
See text 9.4

• After some moderately tedious algebra, (see text
  book Chapter 9, section3) we can derive the
  following equations for the final velocities,

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Example - Elastic Collision
See text 9.4

• Suppose I have 2 identical bumper cars. One is
  motionless and the other is approaching it with
  velocity v1. If they collide elastically, what is
  the final velocity of each car ?
• Note that this means,
• m1 m2 m
• v2B 0

Animation
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Lecture 18, ACT 3Elastic Collisions

• I have a line of 3 bumper cars all touching. A
  fourth car smashes into the others from behind.
  Is it possible to satisfy both conservation of
  energy and momentum if 2 cars are moving after
  the collision?
• All masses are identical, elastic collision.
• A) Yes B) No

Animation
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Example of 2-D elastic collisionsBilliards
See text Ex. 9.11

• If all we know is the initial velocity of the cue
  ball, we dont have enough information to solve
  for the exact paths after the collision. But we
  can learn some useful things...

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Billiards
See text Ex. 9.11

• Consider the case where one ball is initially at
  rest.

pa
pb
vcm
Pa
F
before
after
the final direction of the red ball will depend
on where the balls hit.
See Figure 12-14
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Billiards
See text Ex. 9.11

• We know momentum is conserved pb pa Pa
• We also know that energy is conserved
• Comparing these two equations tells us that
  

pb2 (pa Pa )2 pa2 Pa2 2 pa ? Pa
2
2
2
P
p
p


a
a
b
pa ? Pa 0
and must therefore be orthogonal! Or one
momentum must be zero.
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Billiards
See text Ex. 9.11

• The final directions are separated by 90o.

pa
pb
vcm
Pa
F
before
after
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Lecture 18 ACT 4Pool Shark
See text Ex. 9.11

• Can I sink the red ball without scratching ?
• Ignore spin and friction.

A) Yes B) No C) More info needed
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Lecture 18 ACT 4Pool Shark
See text Ex. 9.11

• From above, after the collision the two balls
  move off at right angles.
• Thus if the red ball goes toward a pocket, so
  does the cue ball

B) No
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Billiards.
See text Ex. 9.11

• More generally, we can sink the red ball without
  sinking the white ball fortunately.

Animation
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Recap of todays lecture

• Momentum and Collisions
• Ch. 9.1-9.4 (part of 9.4)