### Summary

This lesson focuses on the conservation of energy solely between gravitational potential energy and kinetic energy, moving students into the Research and Revise step. Students start out with a virtual laboratory, and then move into the notes and working of problems as a group. A few questions are given as homework. A dry lab focuses on the kinetic and potential energies found on a roller coaster concludes the lesson in the Test Your Mettle phase of the Legacy Cycle.*This engineering curriculum meets Next Generation Science Standards (NGSS).*

### Engineering Connection

Engineers design solutions for our world. Engineers must first master the physical laws that rule our world before designing something that takes advantage of those laws. Different forms of energy are harnessed and used by many different types of engineers. The broader application of this lesson focuses on the transfer of energy within a vehicle and harnessing that energy so it is not lost.

### Pre-Req Knowledge

A working understanding of Newton's laws and flow charting. A firm understanding of how to use Microsoft Excel® or another spreadsheet program before attempting the dry lab (however, this can also be incorporated into the explanation of the lab).

### Learning Objectives

After this lesson, students should be able to:

- Describe the types of mechanical energy.
- List the various forms of potential energy.
- Apply the conservation of energy to problems strictly between gravitational potential energy and kinetic energy.
- Explain why mechanical energy is not always conserved.

### More Curriculum Like This

**It's Tiggerific!**

Students investigate potential energy held within springs (elastic potential energy) as part of the Research and Revise step. Class begins with a video of spring shoes or bungee jumping. Then students move on into notes and problems as a group. A few questions are given as homework. The Test Your Me...

**Physics of Roller Coasters**

Students explore the physics exploited by engineers in designing today's roller coasters, including potential and kinetic energy, friction and gravity. During the associated activity, students design, build and analyze model roller coasters they make using foam tubing and marbles (as the cars).

**Kinetic and Potential Energy of Motion**

Students are introduced to both potential energy and kinetic energy as forms of mechanical energy. A hands-on activity demonstrates how potential energy can change into kinetic energy by swinging a pendulum, illustrating the concept of conservation of energy.

**Swinging Pendulum**

This activity demonstrates how potential energy (PE) can be converted to kinetic energy (KE) and back again. Given a pendulum height, students calculate and predict how fast the pendulum will swing by understanding conservation of energy and using the equations for PE and KE.

###
Educational Standards
Each *TeachEngineering* lesson or activity is correlated to one or more K-12 science,
technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in *TeachEngineering* are collected, maintained and packaged by the *Achievement Standards Network (ASN)*,
a project of *D2L* (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics;
within type by subtype, then by grade, *etc*.

Each *TeachEngineering* lesson or activity is correlated to one or more K-12 science,
technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in *TeachEngineering* are collected, maintained and packaged by the *Achievement Standards Network (ASN)*,
a project of *D2L* (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics;
within type by subtype, then by grade, *etc*.

###### NGSS: Next Generation Science Standards - Science

- Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?

###### Common Core State Standards - Math

- Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?
- Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?

###### International Technology and Engineering Educators Association - Technology

- Energy cannot be created nor destroyed; however, it can be converted from one form to another. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?
- Energy can be grouped into major forms: thermal, radiant, electrical, mechanical, chemical, nuclear, and others. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?

###### National Science Education Standards - Science

- Physical Science (Grades K - 12) Details... View more aligned curriculum... Do you agree with this alignment?

### Introduction/Motivation

Now that we have come up with a few ideas about the hybrid cars, we know that we need to know more about energy. So today we will learn about the two most basic types of energy and how they can be equated to each other. So, let's talk about energy (continue with lecture information provided in the Teacher Background section).

### Lesson Background and Concepts for Teachers

Content Information

Pedagogically, make the first logical step in the conservation of energy between just two forms of mechanical energy. Typically, do this between kinetic energy, which is the energy of motion (KE = ½ mv^{2}), and gravitational potential energy, which is the energy of position (GPE = mgh). However, consider using other forms of energy (that is, thermal, chemical, nuclear, electric, electromagnetic, elastic potential energy and energy carried by longitudinal waves [typically in the form of sound waves or seismic waves]).

The conservation of energy states that the total energy at any two points in time for a close system. For basic purposes, the assumption that we always deal with a closed system can be taken.

E_{1} = E_{2} (1)

Here, we assume that the only forms of energy that can be dealt with are kinetic and gravitational potential energies.

KE_{1} + GPE_{1} = KE_{2} + GPE_{2} (2)

Expanding,

½ mv_{1}
^{2} + mgh_{1} = ½ mv _{2}
^{2} + mgh_{2}

Reducing,

½ v_{1}
^{2} + gh_{1} = ½ v_{2}
^{2 }+ gh_{2} (3)

Therefore, we see that (3) is true regardless of the mass of the object. To start out, problems should be solely KE at one point and GPE at the other. This also requires careful placement of your ground level. So (3) can reduce to:

gh_{1} = ½ v_{2}
^{2} (4)

Only after a few problems, should the case in which we have to include other terms to have (3) be true should be attempted. Just before the second associated activity Energy on a Roller Coaster should the concept of adding other forms of energy be discussed and the law of Conservation of Energy always be expressed as in (1).

Lecture Information

Write the word "energy" on the classroom board. List some of the reasons (from the last lesson) that relate to conservation of energy, and ask: What are some of the forms of energy? Expect students to give answers such as "solar' or "wind," but steer them into the actual names of the forms of energy. Flow chart these responses on the board (which also serves as a lead-in for a later lesson). Circle kinetic and gravitational potential energy, and tell them these two are our focus first.

Choose two student volunteers (pick two who are friends) and pick two lightweight objects from the classroom such as a sponge and a plastic water bottle. Ask the class: Which object will transfer more energy (hurt more) if thrown at a person at the same speed? Listen to the class answers and have the volunteers throw/receive the objects to confirm (this is why friends are chosen). Write on the board that KE *a* mass. Then, ask the class: Which transfers more kinetic energy (hurts more), an object thrown at low speed or the same object at a higher speed? The class answers and the volunteers test and confirm (again, it helps if they are friends in order to avoid any serious throwing). Then place velocity into the equation to have "KE *a* mv." Next, explain that KE is proportional to the square of velocity and that a constant is associated with KE. Then place the ½ and the square into the equation and replace "*a*" with "=" to have "KE = ½ mv^{2}."

Next, determine the equation for gravitational potential energy. Pick two objects from around the room, but this time make sure they have a larger difference in mass (for example, a sponge and the CRC *Handbook of Chemistry and Physics*). Remind students about the storage of mechanical energy. Ask the class: Which object has more energy stored in it if held at the same height. Expect many to answer correctly, and some to not be sure. Explain that we can tell from dropping them and seeing which transfers the most energy to the floor. One way to determine which transfers the most energy is by the loudness of the impact. Expect them to immediately determine the answer, but demonstrate anyway. Then, write on the board, "GPE *a* m." Ask: Which one has more stored energy, a book at 1 m from the floor or 2 m from the floor? Demonstrate this. Then include height in the equation for "GPE *a* mh." Remind students that the attraction of the object to the ground is also determined by something else. Wait until students determine that it must be gravity. Then include g and substitute "=" for "*a*" for "GPE = mgh.

For further background on the physcis presented in this lesson, select the lecture 11 video at: https://archive.org/details/MITclassical_mech. Optionally, you could show students clips from this video, or if you feel appropriate, show students the entire lecture.

At this point, have students conduct the Energy Skate Park associated activity. Assign finishing the virtual lab as homework.

Then work on the classroom board several problems from the text on the conservation of energy between these two forms of energy. Inform students about the real-life presence of friction, as well as what friction does. Next, have students conduct the Energy on a Roller Coaster associated activity.

### Vocabulary/Definitions

conservation of energy: The total amount of energy within a closed system is equal at any two points in time.

friction: Transfer of energy from mechanical energy into other forms.

gravitational potential energy: Energy stored due to position in the vertical dimension.

kinetic energy: Energy of motion.

potential energy: Energy that is stored.

### Associated Activities

- Energy Skate Park - Using an online simulation set at a skate park, students make graph predictions before using the virtual laboratory to create graphs of energy vs. time under different conditions. Students' comprehension of potential and kinetic energy and conservation of energy is strengthened as they predict behavior based on their understanding of these energy concepts and experimentation with the simulation.
- Energy on a Roller Coaster - Students use a roller coaster track to collect position data. They calculate velocity and energy data, and then relate the conversion of potential and kinetic energy to the conversion of energy used in a hybrid car, learning about energy lost to friction.

### Attachments

### Assessment

*Informal Assessment*: While doing problems with the class, make sure to only do the first problem for them. Then solve another with help from the class. Have students individually solve all additional problems, with individual help from the teacher and other students who have had their work checked by the teacher.

*Lab Reports*: Evaluate student lab reports from the virtual lab and dry lab. Have the reports be in a format that the teacher is comfortable with, but that also allows for the checking of the understanding of the concepts by students.

*Quiz*: At lesson end and after both associated activities have been conducted, administer the Conservation of Energy Quiz. Review student answers to gauge their depth of comprehension.

### Contributors

Joel Daniel (funded by the NSF-funded Center for Compact and Efficient Fluid Power at the University of Minnesota); Megan Johnston### Copyright

© 2013 by Regents of the University of Colorado; original © 2006 Vanderbilt University### Supporting Program

VU Bioengineering RET Program, School of Engineering, Vanderbilt University### Acknowledgements

The contents of this digital library curriculum were developed under National Science Foundation RET grant nos. 0338092 and 0742871. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.

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