Physics 0 Lesson 7 Conservation o nergy In this lesson we will learn about one o the ost powerul tools or solving physics probles utilizing the Law o Conservation o nergy. I. Law o Conservation o nergy The Law o Conservation o nergy states that the total aount o energy in a syste reains constant. The energy ay be transored ro one type to another, ro kinetic to potential or kinetic to heat, but the total aount o energy is conserved. For exaple, iagine a skier starting ro rest at the top o the hill at Canada Olypic Park. She has a lot o gravitational potential energy due to the vertical distance ro the top o the hill to the botto. As she slides down the hill, her gravitational potential energy becoes less and less while her kinetic energy (i.e. her speed) becoes larger and larger. I the hill was rictionless, all o her gravitational potential energy ro the top o the hill would be transored into her kinetic energy at the botto. In equation or: (rictionless) kbottoo hill ptopo hill O course, there is a substantial aount o riction between the skis and the snow. Thereore, soe o the initial gravitational potential energy will be converted into heat energy and soe will be converted into kinetic energy. The skier s kinetic energy at the botto o the hill will be the initial gravitational energy inus the heat lost to riction. kbottoo hill ptopo hill heat In these types o probles we are applying the Law o nergy Conservation. This is a very powerul principle or law and we can use it to solve probles which would be quite diicult i we were only using kineatics or dynaics. II. Conservation o energy proble solving The basic ethod or solving probles using the conservation o energy is as ollows: 1. Deterine the dierent ors o energy that are present at the beginning o the proble. I the object is initially higher than it will be at the end o the proble it has gravitational potential energy. I it is in otion it has kinetic energy. I a spring or an elastic is being stretched or copressed, elastic potential energy is involved. I a orce is being applied over a distance then wors being done.. Deterine the dierent ors o energy that are present at the end o the proble. I the object is higher than it was at the beginning it has gravitational potential energy. I it is in otion it has kinetic energy. I a spring or an elastic is being stretched or copressed, elastic potential energy is involved. I a orce is being applied over a distance then wors being done. Dr. Ron Licht 7-1 www.structuredindependentlearning.co
3. Apply the principle o conservation o energy Initial energies = Final energies to the proble. Write a atheatical expression that equates all o the initial ors o energy (gravitational potential, elastic potential, kinetic, work) with all o the inal ors o energy (gravitational potential, elastic potential, kinetic, work, heat, etc.) 4. Substitute in the appropriate equations or each or o energy and solve or the requested value. The ain thing to get used to when solving probles in this way is that you are creating a new atheatical equation or each situation that you encounter. The equation will depend on the context o what is happening in the proble at hand. Be warned, students who try to eorize every type o possible context (there are thousands o possible contexts) end up rustrated and helpless. But students who learn to apply the process/ethod o energy conservation ind the probles easy to solve and work with. xaple 1 A 50 kg object alls 490.5. What is the speed o the object just beore ipact with the ground? The object starts ro a point higher than the ground. I we say that the ground is the zero potential point, the object initially has potential energy ( p i ). In the end it has kinetic energy ( k ). Using conservation o energy we relate the initial energies with the inal energies. Substitute the appropriate equations ro the orula sheet into the relationship. Note that ass cancels out. Rearrange the equation and solve. initial energies = inal energies pi k v gh i v v (9.81 )(490.5) v 98.1 gh i s s Dr. Ron Licht 7 - www.structuredindependentlearning.co
xaple A snowobile driver with a ass o 100 kg traveling at 15.0 /s slas into a snow drit. I the driver sinks 0.50 into the snow drit beore stopping, what is the retarding orce applied by the snow drit? The driver initially has kinetic energy due to his otion ( ). The snow drit applies a orce (F) on the driver through a distance ( d), which eans that work (W = F d) is being done on the driver. In the end he is not oving (no kinetic energy). Using conservation o energy we relate the initial kinetic energy with the work done to stop the driver. Substitute the appropriate equations ro the orula sheet into the relationship. Rearrange the equation and solve. initial energies = inal energies ki W vi Fd vi F d 100kg (15.0 s ) F (0.50) 5 F.5 10 N xaple 3 A 5.0 kg object is thrown vertically down ro the top o a 50.0 tower with a speed o 15.0 /s. What is the speed o the object at the botto o the tower just beore it hits the ground? The object starts ro a point higher than the ground ( p i ) and it has an initial speed ( ). In the end it has kinetic energy ( k ). Using conservation o energy we relate the initial energies with the inal energies. Substitute the appropriate equations ro the orula sheet into the relationship. Note that ass cancels out. Rearrange the equation and solve. initial energies = inal energies pi k vi ghi v v gh v i i v (9.81 )(50.0) (15.0 ) v 34.7 s s s Dr. Ron Licht 7-3 www.structuredindependentlearning.co
xaple 4 A 5.0 gra bullet enters a wooden block at 350 /s and exits the 0 c wide block at 150 /s. What was the orce applied to the bullet by the block? There are at least two ways to conceptualise this proble. First, the bullet has an initial kinetic energy ( ) and a inal kinetic energy ( k ). Work was done on the bullet (i.e. orce through a distance) to slow it down. initial energies = inal energies W W The second way is to use the concept o work (W = k ) that we learned about in Lesson 6. Since the wood is slowing the bullet down the work done is negative. W k k k k W W In any case, we substitute in our equations and solve or the unknown. k W k vi v Fd F (vi v ) d 0.0050kg F (350 s) (0.0) (150 ) 3 F 1.5 10 N s Dr. Ron Licht 7-4 www.structuredindependentlearning.co
xaple 5 A 5 kg object resting at the top o a 15 high inclined plane begins to slide down the plane. At the botto o the plane the object has a speed o 14.0 /s. A. How uch heat energy was produced? The objects initial gravitational potential energy ( p i ) is converted into kinetic ( k ) and heat energy ( h ) due to riction. initial energies = inal energies pi k h h pi k v h ghi 5kg(14.0 s ) h 5kg(9.81 )(15) s 3 h 1.3 10 J B. I the incline is 35.0 long, what is the rictional orce? h h W Fd h F d F 3 1.3 10 J 35.0 F 35.1N xaple 6 A toy car with ass 348 g is pushed up against a copression spring. The spring is copressed by 5.3 c. When the car is released its inal speed is 7.3 /s. What is the spring constant or the copression spring? When the car is pushed up against the spring the car has spring potential energy ( p i ). When it is released it has kinetic energy ( k ). initial energies = inal energies pi k kx v v k x i i 0.348kg(7.3 s ) k (0.053) k 6.6 10 3 N Dr. Ron Licht 7-5 www.structuredindependentlearning.co
III. Practice Probles 1. A otorcycle rider is trying to leap across the canyon as shown in the igure by driving horizontally o the cli. When it leaves the cli, the cycle has a speed o 38.0 /s. Ignoring air resistance, ind the speed with which the cycle strikes the ground on the other side. (46. /s). A 6.00 rope is tied to a tree lib and used as a swing. A person starts ro rest with the rope held in a horizontal orientation, as in the igure. Ignoring riction and air resistance, deterine how ast the person is oving at the lowest point on the circular arc o the swing. (10.8 /s) Dr. Ron Licht 7-6 www.structuredindependentlearning.co
3. One o the astest roller coasters (000 kg) in the world is the Magnu XL - 00 at Cedar Point Parn Sandusky, Ohio. This ride includes an initial vertical drop o 59.3. Assue that the roller coaster has a speed o nearly zero as it crests the top o the hill. A. I the track was rictionless, ind the speed o the roller coaster at the botto o the hill. (34.1 /s) B. The actual speed o the roller coaster at the botto is 3. /s. I the length o tracs 15, what is the average rictional orce acting on the roller coaster? (1.01 x 10 3 N) IV. Hand-in Assignent 1. An 80.0 kg box is pushed up a rictionless incline as shown in the diagra. How uch wors done on the box in oving it to the top? (Hint, think energy, not orces.) (5.49 kj). A 75 g arrow is ired horizontally. The bow string exerts an average orce o 65 N on the arrow over a distance o 0.90. With what speed does the arrow leave the bow string? (39 /s) 3. In the high jup, the kinetic energy o an athlete is transored into gravitational potential energy. With what iniu speed ust the athlete leave the ground in order to lit his center o ass.10 and cross the bar with a speed o 0.80 /s? (6.5 /s) 4. A 50.0 kg pole vaulter running at 10.0 /s vaults over the bar. Assuing that the vaulter's horizontal coponent o velocity over the bar is 1.00 /s and disregarding air resistance, how high was the jup? (5.05 ) 5. I a 4.00 kg board skidding across the loor with an initial speed o 5.50 /s coes to rest, how uch theral energy is produced? (60.5 J) Dr. Ron Licht 7-7 www.structuredindependentlearning.co
6. A roller coaster is shown in the drawing. Assuing no riction, calculate the speed at points B, C, D, assuing it has a speed o 1.80 /s at point A. (4.3 /s, 10.1 /s, 18.9 /s) 7. A water skier lets go o the tow rope upon leaving the end o a jup rap at a speed o 14.0 /s. As the drawing indicates, the skier has a speed o 13.0 /s at the highest point o the jup. Ignoring air resistance, deterine the skier s height H above the top o the rap at the highest point. (1.38 ) 8. A roller coaster vehicle with occupants has a ass o.9 x 10 3 kg. It starts at point A with a speed o 14 /s and slides down the track through a vertical distance o 5 to B. It then clibs in the direction o point C which is 36 above B. An interesting eature o this roller coaster is that due to cost-over-runs and poor planning, the track ends at point C. The occupant is the chie design engineer o the roller coaster ride. stiate the speed o the vehicle at point B and then deterine whether the ellow survives the ride. (6 /s) 9. The speed o a hockey puck (ass = 100.0 g) decreases ro 45.00 /s to 4.68 /s in coasting 16.00 across the ice. a. How uch theral energy was produced? (10.17 J) b. What rictional orce was acting on the puck? (0.6357 N) 10. During an autoobile accident investigation, a police oicer easured the skid arks let by a car (ass = 1500 kg) to be 65 long. I the rictional orce on the car was 7.66 kn during the skid, was the car going aster than the 100 k/h speed liit beore applying the brakes? (slower) 11. A 45.0 kg box initially at rest slides ro the top o a 1.5 long incline. The incline is 5.0 high at the top. I the box reaches the botto o the incline at a speed o 5.0 /s, what is the orce o riction on the box along the incline? (1.3 x 10 N) *1. For the pulley syste illustrated to the right, when the asses are released, what is the inal speed o the 1 kg ass just beore it hits the loor? (7.0 /s) 1 kg 5.0 4.0 kg Dr. Ron Licht 7-8 www.structuredindependentlearning.co
Hot Wheels Activity Proble: What is the relationship between the potential energy o a car at the top o a hill and its kinetic energy at the botto o a hill? How uch heat was lost due to riction? What is the average rictional orce o the track on the car? (Hint: You cannot use conservation o energy to calculate the speed at the botto o the rap. Why?) rap p = g h k = ½ v h (easure) table top d y (easure) d x (easure) Dr. Ron Licht 7-9 www.structuredindependentlearning.co