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Kinetic Energy Lecturer: Professor Stephen T. Thornton

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Reading Quiz: Two marbles, one twice as heavy as the other, are dropped to the ground from the roof of a building. Just before hitting the ground, the heavier marble has A) the same kinetic energy as the lighter one. B) half as much kinetic energy as the lighter one. C) twice as much kinetic energy as the lighter one. D) four times as much kinetic energy as the lighter one.

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Answer: C The velocities will be the same in this case, so the only difference in the kinetic energy is due to the mass. Because the mass is twice as much, the kinetic energy is twice as much.

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Last Time Discussed the concept of work

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Today Discuss kinetic energy Work-energy theorem

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In a baseball game, the catcher stops a 90-mph pitch. What can you say about the work done by the catcher on the ball? A) catcher has done positive work B) catcher has done negative work C) catcher has done zero work Conceptual Quiz

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In a baseball game, the catcher stops a 90-mph pitch. What can you say about the work done by the catcher on the ball? A) catcher has done positive work B) catcher has done negative work C) catcher has done zero work opposite in direction to the displacement of the ball, so the work is negative W = F d cos = 180 º W < 0 The force exerted by the catcher is opposite in direction to the displacement of the ball, so the work is negative. Or using the definition of work (W = F d cos ), because = 180 º, then W < 0. Note that because the work done on the ball is negative, its speed decreases. Conceptual Quiz Follow-up: What about the work done by the ball on the catcher?

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Previous kinematic equation:

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Definition of kinetic energy Unit of kinetic energy is the joule, J. Q. Why do we define this? A. It turns out to make our calculations easier!

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Work-Energy Theorem The total work done on an object is equal to the change in its kinetic energy: This is a general result, even for a force not constant in magnitude and direction. Do spring demo.

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Copyright © 2009 Pearson Education, Inc. Kinetic Energy and the Work-Energy Principle Energy was traditionally defined as the ability to do work. We now know that not all forces are able to do work; however, we are dealing in these chapters with mechanical energy, which does follow this definition.

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Because work and kinetic energy can be equated, they must have the same units: kinetic energy is measured in joules. Energy can be considered as the ability to do work:

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To find work, we have to be sure about what force is exerting the effort. Here we might ask about the work done by friction, gravity, air resistance, or the engine. Not a good figure. Forces are not accurate.

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We need to make sure we can find the work done by every force. (sloppy diagram)

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Doing Work Against Gravity Energy is reclaimed in this case.

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Doing Work Against Friction Energy is not reclaimed in this case.

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Does the Earth do work on the moon? The Moon revolves around the Earth in a nearly circular orbit, with approximately constant tangential speed, kept there by the gravitational force exerted by the Earth. Gravity does no work, because the displacement and force are perpendicular.

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Cable Tension, Work. A 265-kg load is lifted 23.0 m vertically with an acceleration a = 0.150g by a single cable. Determine (a) the tension in the cable; (b) the net work done on the load; (c) the work done by the cable on the load; (d) the work done by gravity on the load; (e) the final speed of the load assuming it started from rest.

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Crate Friction. A 46.0-kg crate, starting from rest, is pulled across a floor with a constant horizontal force of 225 N. For the first 11.0 m the floor is frictionless, and for the next 10.0 m the coefficient of friction is What is the final speed of the crate after being pulled these 21.0 m?

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Paintball. In the game of paintball, players use guns powered by pressurized gas to propel 33-g gel capsules filled with paint at the opposing team. Game rules dictate that a paintball cannot leave the barrel of a gun with a speed greater than 85 m/s. Model the shot by assuming the pressurized gas applies a constant force F to a 33-g capsule over the length of the 32-cm barrel. Determine F (a) using the work-energy principle, and (b) using the kinematic equations and Newton’s second law.

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A child on a skateboard is moving at a speed of 2 m/s. After a force acts on the child, her speed is 3 m/s. What can you say about the work done by the external force on the child? A)positive work was done B) negative work was done C) zero work was done Conceptual Quiz

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A child on a skateboard is moving at a speed of 2 m/s. After a force acts on the child, her speed is 3 m/s. What can you say about the work done by the external force on the child? A)positive work was done B) negative work was done C) zero work was done The kinetic energy of the child increased because her speed increasedincrease in KE positive work being done KE f > KE i work W must be positive The kinetic energy of the child increased because her speed increased. This increase in KE was the result of positive work being done. Or, from the definition of work, because W = KE = KE f – KE i and we know that KE f > KE i in this case, then the work W must be positive. Conceptual Quiz Follow-up: What does it mean for negative work to be done on the child?

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Conceptual Quiz A) 20 m B) 30 m C) 40 m D) 60 m E) 80 m If a car traveling 60 km/hr can brake to a stop within 20 m, what is its stopping distance if it is traveling 120 km/hr? Assume that the braking force is the same in both cases.

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F d = W net = KE = 0 – mv 2, |F| d = mv 2. and thus, |F| d = mv 2. doubles Therefore, if the speed doubles, four times larger the stopping distance gets four times larger. Conceptual Quiz A) 20 m B) 30 m C) 40 m D) 60 m E) 80 m If a car traveling 60 km/hr can brake to a stop within 20 m, what is its stopping distance if it is traveling 120 km/hr? Assume that the braking force is the same in both cases.

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Copyright © 2009 Pearson Education, Inc. Work Done by a Constant Force Solving work problems: 1. Draw a free-body diagram. 2. Choose a coordinate system. 3. Apply Newton’s laws to determine any unknown forces. 4. Find the work done by a specific force. 5. To find the net work, either a)find the net force and then find the work it does, or b)find the work done by each force and add.

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By what factor does the kinetic energy of a car change when its speed is tripled? A) no change at all B) factor of 3 C) factor of 6 D) factor of 9 E) factor of 12 Conceptual Quiz

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By what factor does the kinetic energy of a car change when its speed is tripled? A) no change at all B) factor of 3 C) factor of 6 D) factor of 9 E) factor of 12 mv 2 speed increases by a factor of 3KE will increase by a factor of 9 Because the kinetic energy is mv 2, if the speed increases by a factor of 3, then the KE will increase by a factor of 9. Conceptual Quiz Follow-up: How would you achieve a KE increase of a factor of 2?

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Conceptual Quiz A) quarter as much B) half as much C) the same D) twice as much E) four times as much Two stones, one twice the mass of the other, are dropped from a cliff. Just before hitting the ground, what is the kinetic energy of the heavy stone compared to the light one?

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Consider the work done by gravity to make the stone fall distance d: KE = W net = F d cos KE = mg d greater massgreater Thus, the stone with the greater mass has the greater KEtwice KE, which is twice as big for the heavy stone. Conceptual Quiz A) quarter as much B) half as much C) the same D) twice as much E) four times as much Two stones, one twice the mass of the other, are dropped from a cliff. Just before hitting the ground, what is the kinetic energy of the heavy stone compared to the light one? Follow-up: How do the initial values of gravitational PE compare?

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Conceptual Quiz A) 0 30 mph B) 30 60 mph C) both the same A car starts from rest and accelerates to 30 mph. Later, it gets on a highway and accelerates to 60 mph. Which takes more energy, the 0 30 mph, or the 30 60 mph?

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mv 2 velocity squared The change in KE ( mv 2 ) involves the velocity squared. m (30 2 − 0 2 ) = m (900) So in the first case, we have: m (30 2 − 0 2 ) = m (900) m (60 2 − 30 2 ) = m (2700) In the second case, we have: m (60 2 − 30 2 ) = m (2700) bigger energy changesecond case Thus, the bigger energy change occurs in the second case. Conceptual Quiz A car starts from rest and accelerates to 30 mph. Later, it gets on a highway and accelerates to 60 mph. Which takes more energy, the 0 30 mph, or the 30 60 mph? A) 0 30 mph B) 30 60 mph C) both the same Follow-up: How much energy is required to stop the 60-mph car?

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The work W 0 accelerates a car from 0 to 50 km/hr. How much work is needed to accelerate the car from 50 km/hr to 150 km/hr? Conceptual Quiz A) 2 W 0 B) 3 W 0 C) 6 W 0 D) 8 W 0 E) 9 W 0

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The work W 0 accelerates a car from 0 to 50 km/hr. How much work is needed to accelerate the car from 50 km/hr to 150 km/hr? A) 2 W 0 B) 3 W 0 C) 6 W 0 D) 8 W 0 E) 9 W 0 Let’s call the two speeds v and 3v, for simplicity. We know that the work is given by W = KE = KE f – Ke i. v v 2 Case #1: W 0 = m (v 2 – 0 2 ) = m (v 2 ) 3vv 2 9v 2 v 2 8v 2 Case #2: W = m ((3v) 2 – v 2 ) = m (9v 2 – v 2 ) = m (8v 2 ) = 8 W 0 Conceptual Quiz Follow-up: How much work is required to stop the 150-km/hr car?

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Conceptual Quiz A) m 1 B) m 2 C) they will go the same distance Two blocks of mass m 1 and m 2 (m 1 > m 2 ) slide on a frictionless floor and have the same kinetic energy when they hit a long rough stretch ( > 0), which slows them down to a stop. Which one goes farther? m1m1 m2m2

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same KE same work forcelessm 2 distancegreater With the same KE, both blocks must have the same work done to them by friction. The friction force is less for m 2 so stopping distance must be greater. Conceptual Quiz A) m 1 B) m 2 C) they will go the same distance Two blocks of mass m 1 and m 2 (m 1 > m 2 ) slide on a frictionless floor and have the same kinetic energy when they hit a long rough stretch ( > 0), which slows them down to a stop. Which one goes farther? m1m1 m2m2 Follow-up: Which block has the greater magnitude of acceleration?

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A golfer making a putt gives the ball an initial velocity of v 0, but he has badly misjudged the putt, and the ball only travels one-quarter of the distance to the hole. If the resistance force due to the grass is constant, what speed should he have given the ball (from its original position) in order to make it into the hole? A) 2 v 0 B) 3 v 0 C) 4 v 0 D) 8 v 0 E) 16 v 0 Conceptual Quiz

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A golfer making a putt gives the ball an initial velocity of v 0, but he has badly misjudged the putt, and the ball only travels one-quarter of the distance to the hole. If the resistance force due to the grass is constant, what speed should he have given the ball (from its original position) in order to make it into the hole? A) 2 v 0 B) 3 v 0 C) 4 v 0 D) 8 v 0 E) 16 v 0 four times the distance four times the workinitial KE must be four times greater increase in the initial speed by a factor of 2KE = mv 2 In traveling four times the distance, the resistive force will do four times the work. Thus, the ball’s initial KE must be four times greater in order to just reach the hole—this requires an increase in the initial speed by a factor of 2, because KE = mv 2. Conceptual Quiz

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