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UCSD Physics 10 Momentum & Impulse

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UCSD Physics 10 Winter 20062 Momentum, p The linear momentum p of an object is the product of the object’s mass m and velocity v: Linear momentum is a vector quantity that points in the same direction as the velocity. SI Unit of Linear Momentum: kilogram · meter/second = (kg · m/s)

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UCSD Physics 10 Winter 20063 Impulse, J The impulse J of a force is the product of the average force and the time interval t during which the force acts: Impulse is a vector quantity and has the same direction as the average force. SI Unit of Impulse: newton · second = (N · s)

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UCSD Physics 10 Winter 20064 IMPULSE–MOMENTUM THEOREM When a net force acts on an object, the impulse of the net force is equal to the change in momentum of the object: IMPUSE = CHANGE IN MOMENTUM

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UCSD Physics 10 Winter 20065 Derivation of the Impulse-Momentum theorem Start by applying Newton’s 3 rd Law.Start by applying Newton’s 3 rd Law. F 1 + F 2 = 0; Equal and opposite forces! Expand the Equation!Expand the Equation! m 1 a 1 + m 2 a 2 = 0; but a = Δv/t m 1 Δv 1 /t + m 2 Δv 2 /t = 0; cancel out time. Therefore, m 1 Δv 1 + m 2 Δv 2 = 0Therefore, m 1 Δv 1 + m 2 Δv 2 = 0 Conservation of Linear Momentum.Conservation of Linear Momentum.

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UCSD Physics 10 Winter 20066 Linear Momentum Often misused word, though most have the right ideaOften misused word, though most have the right idea Momentum, denoted p, is mass times velocityMomentum, denoted p, is mass times velocity p = m·v Momentum is a conserved quantity (and a vector)Momentum is a conserved quantity (and a vector) –Often relevant in collisions (watch out for linebackers!)

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UCSD Physics 10 Winter 20067 Momentum is a vector!!! Two Linebackers, as shown, are about to have a collision!!! – Assume they each have 90 kg of mass What is the total momentum of both linebackers!What is the total momentum of both linebackers! p = m·v Momentum is a conserved quantity (and a vector)Momentum is a conserved quantity (and a vector) –What is the total momentum of the linebackers after the collisions?

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UCSD Physics 10 Winter 20068 News headline: Wad of Clay Hits Unsuspecting Sled 1 kg clay ball strikes 5 kg sled at 12 m/s and sticks!!1 kg clay ball strikes 5 kg sled at 12 m/s and sticks!! –Determine the initial momentum of the system before the collision. Momentum before collision: (1 kg)(12 m/s) + (5 kg)(0 m/s) = 12 kg·m/sMomentum before collision: (1 kg)(12 m/s) + (5 kg)(0 m/s) = 12 kg·m/s Momentum is a conserved quantity (and a vector)Momentum is a conserved quantity (and a vector) –What is the final momentum of the system after the collision? Momentum after = 12 kg·m/sMomentum after = 12 kg·m/s –How fast does the total mass of the sled and clay move off together? P = mv 12 kg·m/s = (6 kg)· v ; v = (2 m/s)P = mv 12 kg·m/s = (6 kg)· v ; v = (2 m/s)

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UCSD Physics 10 Winter 20069 Collisions Two types of collisionsTwo types of collisions –Elastic: Energy not dissipated out of kinetic energy Bouncy –Inelastic: Some energy dissipated to other forms Sticky Perfect elasticity unattainable (perpetual motion)Perfect elasticity unattainable (perpetual motion)

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UCSD Physics 10 Winter 200610 Equal and opposite forces during any collision or interaction. Forces are always equal and opposite!!!Forces are always equal and opposite!!! –Newton’s 3 rd Law of Motion! Which Canoe experience the greater acceleration?Which Canoe experience the greater acceleration? How do the momenta of each canoe compare?How do the momenta of each canoe compare? –Be sure to consider both MASS and VELOCITY!!!

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UCSD Physics 10 Winter 200611 Equal and opposite forces, impulses and changes in momentum during a collision Impulse = Product of Force and TimeImpulse = Product of Force and Time –J = FΔt Impulse is a vector in the same direction as the net force. Units of impulse are N∙s = kg∙m/s Change in momentum = Product of mass and Δ velocityChange in momentum = Product of mass and Δ velocity –Δ p = mΔv Change in momentum is a vector. Δ p = m(vf – vi) Units are N∙s = kg∙m/s Impulse = Change in momentum : J = FΔt = Δ p = mΔvImpulse = Change in momentum : J = FΔt = Δ p = mΔv

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UCSD Physics 10 Winter 200612 Concept Check? The magnitude of the initial momentum is 7 kgm/s and the magnitude of the final momentum is - 5 kgm/s. What is the impulse acting on the baseball?The magnitude of the initial momentum is 7 kgm/s and the magnitude of the final momentum is - 5 kgm/s. What is the impulse acting on the baseball? What is the impulse acting on the bat?What is the impulse acting on the bat? Impulse on the ball is – 12 kgm/s FΔt = Δ p = mΔvImpulse on the ball is – 12 kgm/s FΔt = Δ p = mΔv Impulse on the bat is + 12 kgm/s Impulses are equal and opposite!!!Impulse on the bat is + 12 kgm/s Impulses are equal and opposite!!!

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UCSD Physics 10 Winter 200613 Problem - Continued If the bat is in contact with the ball for 0.003 sec, what Avg. force acts on the ball?If the bat is in contact with the ball for 0.003 sec, what Avg. force acts on the ball? J = FΔt, 12kgm/s = F (.003s); F = 4 kNJ = FΔt, 12kgm/s = F (.003s); F = 4 kN What is the rate of change in the momentum?What is the rate of change in the momentum? F = mΔv / t, F = 4 kNF = mΔv / t, F = 4 kN Impulse Graphs!!!Impulse Graphs!!!Graphs

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UCSD Physics 10 Winter 200614 Impulse Graphs The Area under the curve for a force vs time graph represents the impulse or change in momentum of the object.The Area under the curve for a force vs time graph represents the impulse or change in momentum of the object. How would the graph for the baseball appear?How would the graph for the baseball appear? Impulse Graphs!!!Impulse Graphs!!!Graphs

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UCSD Physics 10 Winter 200615 Vertical Jump force-time curve A body weight (BW) line has been added to the graph. The total force acting on the jumper is the ground reaction force minus gravitational force (BW).A body weight (BW) line has been added to the graph. The total force acting on the jumper is the ground reaction force minus gravitational force (BW). The shaded regions of the graph sum to produce the impulse before takeoff and the impulse after landing. The regions below the BW line are "negative areas."The shaded regions of the graph sum to produce the impulse before takeoff and the impulse after landing. The regions below the BW line are "negative areas."

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UCSD Physics 10 Winter 200616 Desk Toy Physics The same principle applies to the suspended-ball desk toy, which eerily “knows” how many balls you let go…The same principle applies to the suspended-ball desk toy, which eerily “knows” how many balls you let go… Only way to simultaneously satisfy energy and momentum conservationOnly way to simultaneously satisfy energy and momentum conservation Relies on balls to all have same massRelies on balls to all have same mass

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UCSD Physics 10 Winter 200617 Collisions Two types of collisionsTwo types of collisions –Elastic: Energy not dissipated out of kinetic energy Bouncy –Inelastic: Some energy dissipated to other forms Sticky Perfect elasticity unattainable (perpetual motion)Perfect elasticity unattainable (perpetual motion)

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UCSD Physics 10 Winter 200618 Elastic Collision: Billiard Balls Whack stationary ball with identical ball moving at velocity v cue 8 To conserve both energy and momentum, cue ball stops dead, and 8-ball takes off with v cue Momentum conservation: mv cue = mv cue, after + mv 8-ball Energy conservation: ½mv 2 cue = ½mv 2 cue, after + ½mv 2 8-ball 8 8 The only way v 0 = v 1 + v 2 and v 2 0 = v 2 1 + v 2 2 is if either v 1 or v 2 is 0. Since cue ball can’t move through 8-ball, cue ball gets stopped.

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UCSD Physics 10 Winter 200619 Inelastic Collision Energy not conserved (absorbed into other paths)Energy not conserved (absorbed into other paths) Non-bouncy: hacky sack, velcro ball, ball of clayNon-bouncy: hacky sack, velcro ball, ball of clay Momentum before = m 1 v initial Momentum after = (m 1 + m 2 )v final = m 1 v initial (because conserved) Energy before = ½m 1 v 2 initial Energy after = ½ (m 1 + m 2 )v 2 final + heat energy

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UCSD Physics 10 Winter 200620 Questions Twin trouble-makers rig a pair of swings to hang from the same hooks, facing each other. They get friends to pull them back (the same distance from the bottom of the swing) and let them go. When they collide in the center, which way do they swing (as a heap), if any? What if Fred was pulled higher than George before release?Twin trouble-makers rig a pair of swings to hang from the same hooks, facing each other. They get friends to pull them back (the same distance from the bottom of the swing) and let them go. When they collide in the center, which way do they swing (as a heap), if any? What if Fred was pulled higher than George before release? A 100 kg ogre clobbers a dainty 50 kg figure skater while trying to learn to ice-skate. If the ogre is moving at 6 m/s before the collision, at what speed will the tangled pile be sliding afterwards?A 100 kg ogre clobbers a dainty 50 kg figure skater while trying to learn to ice-skate. If the ogre is moving at 6 m/s before the collision, at what speed will the tangled pile be sliding afterwards?

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UCSD Physics 10 Winter 200621 Real-World Collisions Is a superball elastic or inelastic?Is a superball elastic or inelastic? –It bounces, so it’s not completely inelastic –It doesn’t return to original height after bounce, so some energy must be lost Superball often bounces 80% original heightSuperball often bounces 80% original height –Golf ball 65% –Tennis ball 55% –Baseball 30% Depends also on surface, which can absorb some of the ball’s energyDepends also on surface, which can absorb some of the ball’s energy –down comforter/mattress or thick mud would absorb

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UCSD Physics 10 Winter 200622 Superball Physics During bounce, if force on/from floor is purely vertical, expect constant horizontal velocityDuring bounce, if force on/from floor is purely vertical, expect constant horizontal velocity –constant velocity in absence of forces –like in picture to upper right BUT, superballs often behave contrary to intuitionBUT, superballs often behave contrary to intuition –back-and-forth motion –boomerang effect

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UCSD Physics 10 Winter 200623 Angular Momentum Another conserved quantity is angular momentum, relating to rotational inertia:Another conserved quantity is angular momentum, relating to rotational inertia: Spinning wheel wants to keep on spinning, stationary wheel wants to keep still (unless acted upon by an external rotational force, or torque)Spinning wheel wants to keep on spinning, stationary wheel wants to keep still (unless acted upon by an external rotational force, or torque) Newton’s laws for linear (straight-line) motion have direct analogs in rotational motionNewton’s laws for linear (straight-line) motion have direct analogs in rotational motion

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UCSD Physics 10 Winter 200624 Angular Momentum Angular momentum is proportional to rotation speed ( ) times rotational inertia (I)Angular momentum is proportional to rotation speed ( ) times rotational inertia (I) Rotational inertia characterized by (mass) (radius) 2 distribution in objectRotational inertia characterized by (mass) (radius) 2 distribution in object

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UCSD Physics 10 Winter 200625 Angular Momentum Conservation Speed up rotation by tucking inSpeed up rotation by tucking in Slow down rotation by stretching outSlow down rotation by stretching out Seen in diving all the timeSeen in diving all the time Figure skaters demonstrate impressivelyFigure skaters demonstrate impressively Effect amplified by moving large masses to vastly different radiiEffect amplified by moving large masses to vastly different radii

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UCSD Physics 10 Winter 200626 Do cats violate physical law? Cats can quickly flip themselves to land on their feet If not rotating before, where do they get their angular momentum? There are ways to accomplish this, by a combination of contortion and varying rotational inertia

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UCSD Physics 10 Winter 200627 For more on falling cats: Websites:Websites: –www.pbs.org/wnet/nature/cats/html/body_falling.htmlwww.pbs.org/wnet/nature/cats/html/body_falling.html play quicktime movie –www.exploratorium.edu/skateboarding/trick_midair_ac tivity.htmlwww.exploratorium.edu/skateboarding/trick_midair_ac tivity.html

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