1. Conservation of Mechanical Energy

2. What do you think?
Imagine two students standing side by side at the top of a water slide. One steps off of the platform, falling directly into the water below. The other student goes down the slide. Assuming the slide is frictionless, which student strikes the water with a greater speed?
Explain your reasoning.
Would your answer change if the slide were not frictionless? If so, how?
When asking students to express their ideas, you might try one of the following methods.
(1) You could ask them to write their answers in their notebook and then discuss them.
(2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups.
The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally.
A variety of answers are possible. Encourage them to explain their ideas so you can better understand the answer they chose. Remind the students that friction is not a consideration. Water slides are not truly frictionless, so they might want to imagine an air hockey table as a slide. Students will find out later in the lesson that, in the absence of friction, the speeds would be the same. The only difference would be the direction each is traveling at impact.

3. What do you think?
What is meant when scientists say a quantity is conserved?
Describe examples of quantities that are conserved.
Are they always conserved? If not, why?
Students may be familiar with conservation of mass as a conservation law. Try to get them to explain what it means for something to be “conserved.” There are many ways to describe conservation but students often struggle. They may say that they know what it means but they can’t figure out how to say it or write it. These questions should be a good exercise for them to express their ideas in writing.
Conservation may be expressed as “no change in the quantity” or “before = after” or “the total always stays the same.”

4. Mechanical Energy (ME)
ME = KE + PEg + PEelastic
Kinetic Energy
Elastic Potential Energy
Gravitational Potential Energy
Hooke’s Law
Discuss ME as a useful tool for studying motion. Do not tell students yet that ME is conserved. They will determine this from the coming slides and calculations.
As an example. toss a ball in the air and talk about the potential energy and kinetic energy as it rises and falls. As another example. show students a pendulum and talk about the PE and the KE changing as it swings.

5. Mechanical Energy (ME)
ME = KE + PEg + PEelastic
Does not include the many other types of energy, such as thermal energy, chemical potential energy, and others
ME is not a new form of energy.
Just a combination of KE and PE
Discuss ME as a useful tool for studying motion. Do not tell students yet that ME is conserved. They will determine this from the coming slides and calculations.
As an example. toss a ball in the air and talk about the potential energy and kinetic energy as it rises and falls. As another example. show students a pendulum and talk about the PE and the KE changing as it swings.

6. Classroom Practice Problems
A 1.00 kg book is dropped from a height of 2.00 m. Assume no air resistance.
Calculate the PE and the KE at the instant the book is released.
Answer: PE = 19.6 J, KE = 0 J
Students should use PE = mgh to get the PE at each point.
To calculate the KE they need to first find the velocity. The easiest way to get v2 is using vf2 = 2gy. (Note that the initial velocity is zero, so it was eliminated from the equation). After getting the velocity, use the equation KE = 1/2 mv2.
After making these calculations, show students the chart on the next slide.

7. Classroom Practice Problems
A 1.00 kg book is dropped from a height of 2.00 m. Assume no air resistance.
Calculate the KE and PE when the book has fallen 1.0 m. (Hint: you will need an equation from Chapter 2.)
Answer: PE = 9.81 J, KE = 9.81 J
Students should use PE = mgh to get the PE at each point.
To calculate the KE they need to first find the velocity. The easiest way to get v2 is using vf2 = 2gy. (Note that the initial velocity is zero, so it was eliminated from the equation). After getting the velocity, use the equation KE = 1/2 mv2.
After making these calculations, show students the chart on the next slide.

8. Classroom Practice Problems
A 1.00 kg book is dropped from a height of 2.00 m. Assume no air resistance.
Calculate the KE and PE just before the book hits the ground
Answer: PE = 0 J, KE = 19.6 J
Students should use PE = mgh to get the PE at each point.
To calculate the KE they need to first find the velocity. The easiest way to get v2 is using vf2 = 2gy. (Note that the initial velocity is zero, so it was eliminated from the equation). After getting the velocity, use the equation KE = 1/2 mv2.
After making these calculations, show students the chart on the next slide.

9. Table of Values for the Falling Book
h (m)
PE(J)
KE(J)
ME(J)
19.6
0.5
14.7
4.9
1.0
9.8
1.5
2.0
Mention the following to the students:
(1) KE and PE change during the fall but ME remains the same (it is conserved).
(2) You might ask students what values would change if air resistance was considered. The KE values would be smaller and the ME would gradually decrease, so ME would not be conserved.
(3) No consideration was given to the path the book took to the floor, only its position. Therefore, the result would be the same even if it slid down a ramp (as long as the ramp was frictionless).

10. Conservation of Mechanical Energy
MEi = MEf
Initial ME = Final ME
The sum of KE and PE remains constant.
One type of energy changes into another type.
For the falling book, the PE of the book changed into KE as it fell.
As a ball rolls up a hill, KE is changed into PE.
Conservation of energy provides an easier method of solving some problems. For example, go back to the table created for the falling book, and point out the fact that they never needed to calculate the velocity. If they know the PE, then the KE is simply (19.6 J - PE).

11. PEmax = mgh
KE = 0
PE = mgh
PE + KE = PEmax
PE = mgh
PE + KE = PEmax
PE = 0
KE = PEmax

12. Conservation of Energy
Acceleration does not have to be constant.
ME is not conserved if friction is present.
If friction is negligible, conservation of ME is reasonably accurate.
A pendulum as it swings back and forth a few times
Previously, in Chapter 2, the equations developed required that acceleration be constant. That is not the case for conservation of ME.
The example with friction provides an opportunity to point out that even though ME may not be conserved, energy in general is conserved. With friction, the material becomes warmer due to the contact between the surfaces, which causes the speed of the vibrating molecules to increase.

13. Conservation of Energy
Consider a child going down a slide with friction.
What happens to the ME as he slides down?
Answer: It is not conserved but, instead, becomes less and less.
What happens to the “lost” energy?
Answer: It is converted into nonmechanical energy (thermal energy).
Previously, in Chapter 2, the equations developed required that acceleration be constant. That is not the case for conservation of ME.
The example with friction provides an opportunity to point out that even though ME may not be conserved, energy in general is conserved. With friction, the material becomes warmer due to the contact between the surfaces, which causes the speed of the vibrating molecules to increase.

14. Classroom Practice Problems
A small 10.0 g ball is held to a slingshot that is stretched 6.0 cm. The spring constant is 2.0  102 N/m.
What is the elastic potential energy of the slingshot before release?
Remind students that J require the use of kg and m (not g and cm).
Answer 1: 0.36 J. Students should use PE = 1/2 kx2 to determine the energy stored in the rubber band.
Answer 2: 0.36 J. Students should use conservation of mechanical energy. PE lost = KE gained
Answer 3: 8.5 m/s. Students should use the KE from the last question and KE = 1/2 mv2 to determine v
Answer 4: 3.7 m. Students may revert to equations from Chapter 2. but this is most easily solved using conservation of ME. They only need to find the height required for KE lost = Peg gained = 0.36 J = mgh.
Ans: 0.36 J

15. Classroom Practice Problems
A small 10.0 g ball is held to a slingshot that is stretched 6.0 cm. The spring constant is 2.0  102 N/m.
What is the kinetic energy of the ball right after the slingshot is released?
Remind students that J require the use of kg and m (not g and cm).
Answer 1: 0.36 J. Students should use PE = 1/2 kx2 to determine the energy stored in the rubber band.
Answer 2: 0.36 J. Students should use conservation of mechanical energy. PE lost = KE gained
Answer 3: 8.5 m/s. Students should use the KE from the last question and KE = 1/2 mv2 to determine v
Answer 4: 3.7 m. Students may revert to equations from Chapter 2. but this is most easily solved using conservation of ME. They only need to find the height required for KE lost = Peg gained = 0.36 J = mgh.
Ans: 0.36 J

16. Classroom Practice Problems
A small 10.0 g ball is held to a slingshot that is stretched 6.0 cm. The spring constant is 2.0  102 N/m.
What is the ball’s speed at the instant it leaves the slingshot?
Remind students that J require the use of kg and m (not g and cm).
Answer 1: 0.36 J. Students should use PE = 1/2 kx2 to determine the energy stored in the rubber band.
Answer 2: 0.36 J. Students should use conservation of mechanical energy. PE lost = KE gained
Answer 3: 8.5 m/s. Students should use the KE from the last question and KE = 1/2 mv2 to determine v
Answer 4: 3.7 m. Students may revert to equations from Chapter 2. but this is most easily solved using conservation of ME. They only need to find the height required for KE lost = Peg gained = 0.36 J = mgh.
Ans: 8.5 m/s

17. Classroom Practice Problems
A small 10.0 g ball is held to a slingshot that is stretched 6.0 cm. The spring constant is 2.0  102 N/m.
How high does the ball rise if it is shot directly upward?
Remind students that J require the use of kg and m (not g and cm).
Answer 1: 0.36 J. Students should use PE = 1/2 kx2 to determine the energy stored in the rubber band.
Answer 2: 0.36 J. Students should use conservation of mechanical energy. PE lost = KE gained
Answer 3: 8.5 m/s. Students should use the KE from the last question and KE = 1/2 mv2 to determine v
Answer 4: 3.7 m. Students may revert to equations from Chapter 2. but this is most easily solved using conservation of ME. They only need to find the height required for KE lost = Peg gained = 0.36 J = mgh.
Ans: 3.7 m

18. Problem Solving Process
Draw a sketch
Identify: KEi, PEi, KEf, and PEf
Include both gravitational and elastic PE, as needed
Apply the equation: (KE + PE)i = (KE + PE)f
Solve for required quantity
Mass, speed, distance, height, etc.

19. Practice Problem
Starting from rest, a child zooms down a frictionless slide from an initial height of 3.00m. What is her speed at the bottom of the slide? Assume she has a mass of 25.0 kg.

20. Nonconservative Forces with Energy Considerations
When nonconservative forces are present, the total mechanical energy of the system is not constant
The work done by all nonconservative forces acting on parts of a system equals the change in the mechanical energy of the system

21. Nonconservative Forces and Energy
In equation form:
The energy can either cross a boundary or the energy is transformed into a form of non-mechanical energy such as thermal energy

22. Transferring Energy By Work By applying a force
Produces a displacement of the system

23. Transferring Energy Heat
The process of transferring heat by collisions between molecules
For example, the spoon becomes hot because some of the KE of the molecules in the coffee is transferred to the molecules of the spoon as internal energy

24. Transferring Energy Mechanical Waves
A disturbance propagates through a medium
Examples include sound, water, seismic

25. Transferring Energy Electrical transmission
Transfer by means of electrical current
This is how energy enters any electrical device

26. Transferring Energy Electromagnetic radiation
Any form of electromagnetic waves
Light, microwaves, radio waves

27. Notes About Conservation of Energy
We can neither create nor destroy energy
Another way of saying energy is conserved
If the total energy of the system does not remain constant, the energy must have crossed the boundary by some mechanism
Applies to areas other than physics

28. Now what do you think?
Imagine two students standing side by side at the top of a water slide. One steps off of the platform, falling directly into the water below. The other student goes down the slide. Assuming the slide is frictionless, which student strikes the water with a greater speed?
Explain your reasoning.
Would your answer change if the slide were not frictionless? If so, how?
Both strike at the same speed. One hits the water vertically, while the other slides in nearly horizontally. The PE lost is the same for both and, therefore, the KE gained is the same as well.
With friction, the student falling straight down would be moving faster, because energy is not conserved for the student on the slide. Some of the lost PE is converted into thermal energy

29. Now what do you think?
What is meant when scientists say a quantity is “conserved”?
Describe examples of quantities that are conserved.
Are they always conserved? If not, why?
Conserved means that the quantity does not change. The quantity neither increases nor decreases. Common examples are the conservation of mass and the conservation of energy. Mechanical energy is considered to be conserved in cases where friction is negligible. If friction is significant, mechanical energy is not conserved.