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Momentum

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Calvin & Hobbes by Bill Watterson

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Newton’s 2 nd Law Reprise l Newton’s second law l can be written: Impulse = Change in momentum Impulse = Change in momentum =

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Momentum & Impulse l Momentum: ; kg-m/s l Impulse: ; N-s l Show that N-s are equivalent to kg-m/s

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Momentum & Impulse l Momentum: ; kg-m/s l Impulse: ; N-s l Show that N-s are equivalent to kg-m/s

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Golf Ball Momentum l Mass of golf ball = kg l Speed of golf ball leaving tee on drive = 70 m/s l What is the momentum of the golf ball? l If the golf club is in contact with the ball for 0.5 ms, what is the average force exerted on the ball by the club?

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Golf Ball Momentum l Mass of golf ball = kg l Speed of golf ball leaving tee on drive = 70 m/s l momentum = (mass)(velocity) l P = ( kg)(70 m/s) = 3.21 kg-m/s

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Force on Golf Ball l Mass of golf ball = kg l Speed of golf ball leaving tee on drive = 70 m/s l P = 3.21 kg-m/s l P = 3.21 kg-m/s – 0 = 3.21 kg-m/s l If the golf club is in contact with the ball for 0.5 ms, what is the average force exerted on the ball by the club?

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Decreasing Momentum Over a Long Time l Hitting dashboard compared to air bag l Landing stiff-legged compared to bending knees l Wooden floor compared to tile l Catching a baseball l P is fixed value l To minimize F, increase t

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Bouncing l Which undergoes the greatest change in momentum: (1) a baseball that is caught, (2) a baseball that is thrown, or (3) a baseball that is caught and then thrown back, if the baseballs have the same speed just before being caught and just after being thrown?

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Bouncing l (3) because it has twice the momentum change of either (1) or (2)

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Momentum Conservation l Newton’s second law can be written: F l It tells us that if F EXT = 0, the total momentum of the system does not change. çThe total momentum of a system is conserved if there are no external forces acting. çIn physics, when we say a quantity is conserved, we mean that it remains constant.

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Momentum Conservation l The concept of momentum conservation is one of the most fundamental principles in physics. l This is a component (vector) equation. çWe can apply it to any direction in which there is no external force applied. l You will see that we often have momentum conservation even when energy is not conserved.

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Momentum Conservation l if mass of each astronaut is 60 kg, and the astronaut on the left is initially moving at 5 m/s, how fast does the pair move after the collision?

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Momentum Conservation l if mass of each astronaut is 60 kg, and the astronaut on the left is initially moving at 5 m/s, how fast does the pair move after the collision? l mv = (2m)v’ l (60 kg)*(5 m/s) = (120 kg)*v’ l v’ = (300/120) = 2.5 m/s

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Big Fish Little Fish

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Big Fish Little fish 2

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Brick on Cart

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Elastic vs. Inelastic Collisions l A collision is said to be elastic when colliding objects rebound without lasting deformation or generation of heat. l Carts colliding with a spring in between, billiard balls, etc. vvivvi l A collision is said to be inelastic when the colliding objects become distorted, generate heat, and possibly stick together çCar crashes, collisions where objects stick together, etc.

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Momentum Conservation l Two balls of equal mass are thrown horizontally with the same initial velocity. They hit identical stationary boxes resting on a frictionless horizontal surface. l The ball hitting box 1 bounces back, while the ball hitting box 2 gets stuck. çWhich box ends up moving faster? (a) (b) (c) (a) Box 1 (b) Box 2 (c) same 1 2

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Momentum Conservation l Since the total external force in the x-direction is zero, momentum is conserved along the x-axis. l In both cases the initial momentum is the same (mv of ball). l In case 1 the ball has negative momentum after the collision, hence the box must have more positive momentum if the total is to be conserved. l The speed of the box in case 1 is biggest! 1 2 x V1V1 V2V2

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Momentum Conservation 1 2 x V1V1 V2V2 mv init = MV 1 - mv fin V 1 = (mv init + mv fin ) / M mv init = (M+m)V 2 V 2 = mv init / (M+m) V 1 numerator is bigger and its denominator is smaller than that of V 2. V 1 > V 2

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Inelastic collision in 2-D l Consider a collision in 2-D (cars crashing at a slippery intersection...no friction). vv1vv1 vv2vv2 V beforeafter m1m1 m2m2 m 1 + m 2

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Inelastic collision in 2-D... l We can see the same thing using vectors: P pp1pp1 pp2pp2 P pp1pp1 pp2pp2

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Explosion (inelastic un-collision) Before the explosion: M m1m1 m2m2 v1v1 v2v2 After the explosion:

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Explosion... P l No external forces, so P is conserved. P l Initially: P = 0 Pvv l Finally: P = m 1 v 1 + m 2 v 2 = 0 vv m 1 v 1 = - m 2 v 2 M m1m1 m2m2 vv1vv1 vv2vv2

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