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Chapter 10: Linear Momentum & Collisions DEHS Physics 1

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Linear Momentum Linear momentum is the product of an object’s mass and velocity It is a vector quantity, it’s direction matters! Units of kg. m/s For a system of objects, you can find the total momentum by a simple vector sum

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Momentum and N2L Newton’s 2 nd Law as we have expressed it so far: is only valid when objects have constant mass The more general law is This is valid even if the mass varies

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Example 10-1 An apple that weighs 2.7 N falls vertically downward from rest for 1.4 s. (a) What is the change in the apple’s momentum per second? (b) What is the total change in the apple’s momentum during the 1.4 s fall?

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Impulse The impulse given to an object is the product of the average force and the length of time that force acts Momentum-Impulse Theorem states that the impulse acting on a object is equivalent to the change in the object’s momentum

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Baseball Physics When a baseball or softball is hit, an enormous force (thousands of Newtons) acts for a very short time

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Example 10-2 A kg baseball is moving toward home plate with a speed of 43.0 m/s when it is bunted (hit softly). The bat exerts an average force of 6.50×10 3 N on the ball for 1.30 ms. The average force is directed toward the pitcher, which we take to be the positive x direction. (a) Calculate the impulse acting on the ball. (b) What is the final speed of the ball?

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Example 10-3 (conceptual) A person stands under an umbrella during a rain shower. A few minutes later the raindrops turn to hail – though the number of “drops” hitting the umbrella per time and their speed remain the same. Is the force required to hold the umbrella in the hail more than, less than, or the same as the force required in the rain?

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Example 10-4 After winning a prize on a gameshow, a 72-kg contestant jumps for joy. (a) If the jump results in an upward speed of 2.1 m/s what is the impulse experienced by the contestant? (b) Before the jump, the floor exerts an upward force of mg on the contestant. What additional average upward force does the floor exert if the contestant pushes down on it of 0.36 s?

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Internal vs External Forces Internal forces are forces that act between two objects within the system – They always act in action-reaction pairs – They are always equal in magnitude and opposite in direction so they always sum to zero Internal forces have absolutely no effect on the net momentum of the system

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Internal vs External Forces The net external force acting on a system will exert an impulse on the system If the net external force acting on a system is zero, the change in the net momentum will be zero (p net will be constant)

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Conservation of Linear Momentum The linear momentum of a system is always conserved (as long as no external force act on the system) This does not mean that the momentum of a specific object remains constant, only that the sum of all object’s momenta remains constant

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Expand out the formula: 3 situations

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Example 10-5 Two groups of canoeists meet in the middle of a lake. After a brief visit, a person in a 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, and the mass of canoe 2 and its occupants is 250 kg, find the momentum of each canoe after 1.20 s of pushing.

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Collisions A collision is a situation in which two objects strike one another, exerting a relatively strong force for a relatively short period of time – Net external force must be zero or negligibly small There are elastic collisions in which the mechanical energy of the system is conserved There are inelastic collisions in which some of the initial mechanical energy of the system is converted to nonmechanical energy (sound, heat)

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Inelastic Collisions In an inelastic collision some of the system’s mechanical energy is lost The momentum is conserved: But the KE is not: When objects stick together after colliding, it is called a completely inelastic collision (aka perfectly inelastic)

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Example 10-6 On a touchdown attempt, a 95.0 kg running back runs toward the end zone at 3.75 m/s. A 111-kg linebacker moving at 4.10 m/s meets the runner in a head-on collision. If the two players stick together, (a) what is their velocity immediately after the collision? (b) What are the initial and final kinetic energies of the system?

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Example 10-7 In a ballistic pendulum, a bullet of mass kg is fired with an initial speed v i at the bob of a pendulum. The bob has a mass 1.5 kg, and is suspended by a rod of negligible mass and length 0.75 m. After the collision, the bullet and the bob stick together and swing through an arc, displacing the rod by an angle 42°. Find the initial speed of the bullet. Step 1: Find the speed after the collision using C of M Step 2: Find the height of the bullet & bob using C of E Step 3: Find the height in terms of L & θ

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Example 10-8 A bullet of mass m = 75 g is fired with an initial velocity of 450 m/s at a wooden block of mass M = 2.25 kg that is at rest on a level surface. It emerges on the other side of the block traveling with a reduced speed of 125 m/s. The coefficient of kinetic friction between the block and the surface is How far will the block slide before it comes to rest?

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Elastic Collisions In an elastic collision the system’s mechanical The momentum is conserved: And so is the KE: You are going to have to use Conservation of Momentum AND Conservation of Energy to solve problems

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Example 10-9 The “Newton’s Cradle” is a common tabletop toy that demonstrates some of the basic principles of elastic collisions. If you drop one ball on a side, one ball pops out on the other side; you drop two balls on a side and two ball pop out on the other side. Your friend says, making a clever observation, “The conservation of momentum wouldn’t be violated if I dropped two balls on one side and then only one ball pops out on the other side with twice the speed.” Explain to your friend why this doesn’t occur.

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Some solved results… Suppose two objects of masses m 1 and m 2 undergo an elastic 1D collision, where m 1 is moving with speed v 0 and m 2 is at rest before the collision C of M says: C of E says: Solving (with very messy algebra)

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Unique Results Case 1, where m 1 >> m 2 like a bowling ball colliding with a ping-pong ball Case 2, where m 1 << m 2 like a ping-pong ball colliding with a bowling ball Case 3, where m 1 = m 2 like two pool balls colliding

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Example A fly is happily maintaining a fixed position about 10 ft above the ground when an elephant charges out of the brush and collides with it. The fly bounces elastically off the forehead of the elephant. If the initial speed of the elephant was 4.5 m/s, what are the speeds of the elephant and the fly after the collision?

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Even More General Solutions For two objects that collide in a 1D ELASTIC collision, the solutions have been worked out: NOTE: These are sign sensitive (the v’s are velocity and thus have a direction that is indicated by the sign)

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Example The three air carts are shown below have masses, reading from left to right of 4m, 2m, and m, respectively. The most massive cart has an initial speed of 5 m/s; the other two carts are at rest initially. All carts are equipped with spring bumpers that give elastic collisions. (a) Find the final speed of each cart. (b) verify that the final KE of the system is equal to the initial KE of the system.

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Example A pair of bumper cars in an amusement park ride collide elastically as one approaches the other directly from the rear. One has a mass of 450 kg and the other 550 kg, due to differences in passenger mass. If the lighter one approaches at 4.50 m/s and the other is moving at 3.70 m/s. Calculate (a) their velocities after the collision. (b) the change in momentum of each.

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Example Your physics teacher is likely to show you the following experiment: A tennisball is stacked on top of a basketball and both are dropped at the same time. The two bounce off the ground and then the tennisball flies into the air several times higher than the basketball. Assume that all collisions are completely elastic and that the mass of the basketball is significantly larger than the mass of the tennisball. If the basketball rises to a height h above the floor, find the height that the tennisball rises to.

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Even More General Solutions For two objects that collide in a 1D ELASTIC collision, their relative velocities remains constant and so: NOTE: Again, these are sign sensitive (the v’s are velocity and thus have a direction that is indicated by the sign)

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Example Using a “gravitational slingshot”, spacecraft can gain a significant increase in speed. One such attempt at this maneuver occurred soon before the Cassini probe, headed to Saturn, performed a “slingshot” off Jupiter. Assuming that the speed of the probe before that maneuver is v i = 10.4 km/s and the Jupiter’s speed u = 9.6 km/s is essentially unchanged, find the speed of the probe following the “slingshot”.

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