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 Chapter 20
 Dr. Steve Rapp

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 Define momentum and impulse; recognize the connection between Newton’s
second law and the impulsemomentum equation; be able to calculate
changes in momentum.
 Recognize the connection between Newton’s third law of motion and
conservation of momentum; define a closed, isolated system

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 State the law of conservation of momentum; recognize its implications,
especially in collision problems.
 Distinguish between external and internal forces and use this
distinction in solving problems.

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 Explain the extension of the law of conservation of momentum to two or
three dimensions; apply vectors to the
solution of momentum conservation problems.

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 The momentum is the product of the mass of an object and its velocity: p
= mv
 Using Newton’s second law (F = ma) we can come up with the following:
 acceleration = v/ t; F = ma; F= m v/
t and therefore: F t = m v.

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 The product of a force and the time interval during which it is applied
(F t) is called impulse.
 Impulse is the change in momentum: J = F t where F is in newtons, t is in
seconds. The unit for impulse is
therefore N s.

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 What is the momentum of a 15 kg bicycle moving at 12 m/s?
 p=mv = 15 kg (12 m/s) = 180 kg m/s
 A bullet with a mass of .0025 kg has a momentum of 1.8 kg m/s. What is the magnitude of its velocity?
 v= p/m = 1.8 kg m/s = 720 m/s

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 A gust of wind exerts a 300 N net force on a 100 kg sailboat for 15
sec. What is the change in
momentum and velocity of the boat?
 J = F t = 300 N (15 s) = 4500
N s
 J = p and p = mv: therefore
 v = p/m = 4500 N s = 4.5 m/s

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 When no net forces are acting on a system of objects the total vector
momentum of the system remains constant.
 Newton’s third law is a special application of the conservation of
momentum.

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 Collisions in which the objects stick together.
 Two carts A & B are traveling toward one another along the same
straight line (collision in one dimension) and have an inelastic
collision.
 The diagram on the next screen illustrates this.

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 Carts are moving toward each other.

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 The carts are of equal mass and speeds (velocities equal in magnitude)
 Momentum of cart A = m_{a}v_{a}
 Momentum of cart B = m_{b}v_{b}
 Since the direction of travel of
carts A & B is opposite
v_{a} = v_{b}
 Therefore, m_{a}v_{a} = m_{b}v_{b }and
 m_{a}v_{a} + m_{b}v_{b }= 0
 Total vector momentum = 0

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 Objects rebound without loss of kinetic energy (perfectly elastic
collisions).
 When objects collide at different angles they are said to collide in 2
dimensions.

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 Ball R at rest is hit by ball B

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 A vector diagram shows the momenta of the balls before and after
collision.

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 A ball having a mass of .25 kg and a velocity of .20 m/s eastward
collides with another ball having a mass of .10 kg and a velocity of .10
m/s and along the same straight line.
After the collision the more massive ball has a velocity of .14
m/s eastward. What is the
velocity of the less massive ball?

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 m = mass of more massive ball
 m’ = mass of least massive ball
 Vi & Vf = initial and final
velocity of more massive ball
 V’i & V’f = initial & final velocity of least massive ball
 Remember: total vector momentum after the collision = that before the
collision.

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 mVi + m’V’i = mVf + m’V’f
 V’f = mVi + m’V’i  mVf

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 Two balls with equal mass collide.
Before the collision, ball b moves with a speed of .80 m/s. After the collision, the balls are
observed to move along paths that are perpendicular to each other. If the angle between mrV’r and mbVb in
the vector diagram is 40^{o}, calculate the speed of each ball
after the collision.

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 Vector diagram showing
 collision.

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 Since the masses are equal we can eliminate them from our problem.
 cos 40 = V’r/Vb sin 40 = V’b/Vb
 V’r = cos 40 (.80 m/s) V’b = sin 40 (.80)
 V’r = .77 (.80 m/s) V’b =.64(.80 m/s)
 V’r = .62 m/s V’b = .51 m/s

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 Momentum = mass x velocity
 Change in momentum = impulse
 Impulse = force x time interval
 Newton’s 3rd law is another statement of the law of conservation of
momentum.
 Law of Conservation of Momentum states: In a closed, isolated system,
the total momentum of the system does not change.
 The momentum of a system changes when a net external force acts on it.

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