Dr. Jie Zou PHY 1151G Department of Physics

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Chapter 9

Linear Momentum and Collisions

Dr. Jie Zou PHY 1151G Department of Physics

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Linear Momentum Definition of Linear Momentum,

Linear momentum p is defined as the product of the mass m and velocity v of an object.

SI unit: kg·m/s. Example: A 1180-kg car drives along a city

street at 30.0 mi/h (13.4 m/s). What is the magnitude of the car’s momentum? Using p = mv, we find p = (1800 kg)(13.4 m/s) =

15,800 kg·m/s.

Dr. Jie Zou PHY 1151G Department of Physics

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Impulse Definition of impulse I:

Impulse is defined to be the average force Fav times the length of application time, t,

SI unit: N·s = (kg·m/s2).s = kg·m/s, the same unit as the units of momentum.

Impulse is the change in momentum: I = Fav∆t = ∆p

Dr. Jie Zou PHY 1151G Department of Physics

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Example A batter hits a ball, sending it back

toward the mound. The mass of the baseball is 0.145 kg. The ball was approaching the batter with a speed of 90.0 mi/h and headed toward the pitcher at 60 mi/h after the batter hit the ball. If the ball and bat were in contact for 1.20 ms, what is the average force exerted by the bat?

Dr. Jie Zou PHY 1151G Department of Physics

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p 1,f

p 2,f

p 3,f ...

p 1,i

p 2,i

p 3,i ...

Conservation of Linear Momentum Conservation of momentum for a system of

objects: If the net external force acting on a system is zero, its net (total) momentum is conserved. That is, It is important to note that this statement applies only to

the net momentum of a system, not to the momentum of each individual object.

Internal and external forces: Internal forces: forces acting between objects within the

system. External forces: forces applied from outside the system.

Dr. Jie Zou PHY 1151G Department of Physics

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Example: Conservation of Momentum

Two groups of canoeists meet in the middle of a lake. After a brief visit, a person in canoe 1 pushes on canoe 2 with a force of 46 N to separate the canoes. If the mass of canoe 1 and its occupants is 130 kg, and the mass of canoe 2 and its occupants is 250 kg, find the momentum of each canoe after 1.20 s of pushing (neglect water resistance).

Dr. Jie Zou PHY 1151G Department of Physics

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Example 2: Conservation of Momentum

When a bullet is fired from a rifle, the forces present in the horizontal direction are internal forces.

The total momentum of the rifle-bullet system in the horizontal direction is conserved.

The momentum of the system of the rifle and the bullet before firing is zero. After firing, the net momentum of the system is still zero.

Dr. Jie Zou PHY 1151G Department of Physics

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Collisions Collisions: By a collision we mean a

situation in which two objects strike one another, and in which the net external force is either zero or negligibly small. During a collision, the total momentum of a

system is conserved. The system’s kinetic energy is not

necessarily conserved.

Dr. Jie Zou PHY 1151G Department of Physics

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Inelastic and Elastic Collisions Elastic collisions: after a collision, the final

kinetic energy of the system is equal to the initial kinetic energy, Kf = Ki.

Inelastic collisions: collisions in which the kinetic energy is not conserved, Kf Ki. Completely inelastic collisions: When objects

stick together after colliding, the collision is completely inelastic.

Dr. Jie Zou PHY 1151G Department of Physics

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Example: Completely Inelastic Collisions

Ballistic pendulum: In a ballistic pendulum, an object of mass m is fired with an initial speed v0 at the bob of a pendulum. The bob has a mass M, and it suspended by a rod of negligible mass. After the collision, the object and the bob stick together and swing through an arc, eventually gaining a height h. Find the height h in terms of m, M, v0 and g.

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Example: Elastic Collisions

Consider a head-on collisions of two carts on an air track. The carts are provided with bumpers that give an elastic bounce when the carts collide. Let’s suppose that initially cart 1 is moving to the right with a speed v0 toward cart 2, which is at rest. If the masses of the carts are m1 and m2, respectively, what will be the speed of cart 1 and cart 2 after the collisions?

Dr. Jie Zou PHY 1151G Department of Physics

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In the previous example, both momentum and kinetic energy of the system are conserved: Momentum conservation: m1v0 = m1v1,f + m2v2, f

Kinetic energy conservation: (1/2)m1v02 = (1/2)m1v1,f

2 + (1/2)m2v2,f

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Algebra yields the following results:

v1,f m1 m2

m1 m2

v0, and v2,f

2m1

m1 m2

v0

Derivation

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Elastic collisions in 2D An example: Consider the

collision of two 7.00-kg curling stones. One stone is at rest initially, the other approaches with a speed v1,i = 1.50 m/s. The collision is not head-on, and after the collision, stone 1 moves with a speed of v1,f = 0.610 m/s in a direction 66.0 away from the initial line of motion. What is the speed and direction of stone 2?

Answer: v2,f = 1.37 m/s in a direction of 24.0

Dr. Jie Zou PHY 1151G Department of Physics

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Homework Chapter 9, Page 267, Problems: # 2,

11, 17, 28, 31.