Conservation of Energy and Momentum in Collisions

From wiki.NGSciEd.org

ISCI 6501: Staci Babykin and Jason Steele

Distortion in a fluid. Image Credit: Professor M. Kostic, Northern Illinois University[1]

Grade Level and GPS

Grade Level Targeted: High School Physics

Science GPS Targeted:

• SCSh1: Students will evaluate the importance of curiosity, honesty, openness, and skepticism in science.

a) Exhibit the above traits in their own scientific activities.

b) Recognize that different explanations often can be given for the same evidence.

c) Explain that further understanding of scientific problems relies on the design and execution of new experiments which may reinforce or weaken opposing explanations.

• SP3: Students will evaluate the forms and transformations of energy.

a) Analyze, evaluate, and apply the principle of conservation of energy and measure the components of work-energy theorem by

• describing total energy in a closed system
• identifying different types of potential energy
• calculating kinetic energy given mass and velocity
• relating transformations between potential and kinetic energy

c) Measure and calculate the vector nature of momentum.

d) Compare and contrast elastic and inelastic collisions.

e) Demonstrate the factors required to produce a change in momentum.

Content

Background Information

Conservation of Momentum

The collision of two objects can be described in terms of momentum and energy conservation. The law of conservation of linear momentum is a fundamental law of physics and states that the total momentum of a closed system is constant. This means that in a system of objects which has no interactions with external agents, the momentum is always constant. Because position in space is a vector quantity, momentum is also a vector; therefore, when considering the momentum of a system, the total momentum is the vector sum of the momenta of the objects and must have both a magnitude and a direction. Conservation of momentum says that if an object is at rest at the beginning of the time period, then the initial momentum is zero and the final momentum must also be zero.

The formula for momentum is: p = m * v where p is momentum (in kg*m/s), m is mass (in kg), and v is velocity (in m/s).

Conservation of linear momentum can be represented by the following formula: where u 1 and u 2 are the velocities before the collision and v 1 and v 2 are the velocities after the collision.

A Newton's cradle demonstrates the law of conservation of momentum. Animation credit: wikipedia [2]

Conservation of Energy

In contrast to the conservation of linear momentum, the law of conservation of energy takes a slightly different look. The law of conservation of energy states that the total amount of energy in a closed system remains constant; in other words, energy can be neither created nor destroyed, but can only change form. While the total amount of energy in the system remains constant, the kinetic and potential energy are not always conserved in the system--it depends on the type of collision. While momentum is always conserved, kinetic energy is not always conserved.

There are three basic types of collisions that can be analyzed:

• Elastic: kinetic energy is conserved
• Inelastic: a portion of the kinetic energy is transformed
• Completely (or perfectly) inelastic: the objects stick together after collision; all the kinetic energy is transferred

The majority of collisions in the real world are inelastic collisions because some kinetic energy is transferred or transformed in the collision.

Kinetic energy can be determined from the equation: .

Integration of Science

The objective of this lab is to show tangibly the effects of the different types of collisions on conservation of kinetic energy and momentum. This lab requires students to be able to calculate momentum and kinetic energy from the graph produced by the collision of two carts. By using the graphical data, students can analyze the momentum and kinetic energy pre- and post-collision.

Teaching Considerations

Common Misconceptions Addressed

Students often think that since energy is always conserved, kinetic energy is also always conserved in all situations. This lab illustrates the important concept that while energy is always conserved in the system, it is total energy not just kinetic energy that must be considered.

Students also frequently misunderstand the significance of the completely inelastic collision: momentum is always conserved and is conserved in that collision as well, but kinetic energy is completely transformed.

Materials Necessary

This lab, as demonstrated, uses motion detectors and graphing software to obtain data. However, it could be adapted to work without that equipment.

Inquiry Lab

• Before beginning the lab, have students answer the following questions about the collision between two billiard balls:

a) Consider a head-on collision between two billiard balls. One is initially at rest and the other moves toward it. Sketch a position vs. time graph for each ball, starting with time before the collision and ending a short time after.

b) Is momentum conserved in the collision? In kinetic energy? Explain.

• Explain the general procedure for collecting data:

Start collecting data using the motion detectors, then gently push one cart towards the other, stationary, cart.

After performing the collision, from the velocity vs time graph, you can determine the velocity during time intervals before and after the collision.

Highlight the time interval to be analyzed, then click the statistics button to read the average values. Determine the average velocity for cart 1 and cart 2 before and after the collision.

All the data collected needs to be kept in a data table with columns for: run number, initial velocity for cart 1 and cart 2, and final velocity for cart 1 and cart 2.

Data should be collected 3 times for each set of bumpers:

Magnet-Magnet

Magnet-Velcro

Velcro-Velcro

• Data Analysis

Create a second data table to help analyze the data with columns for: run number, initial momentum for carts 1 and 2, final momentum for carts 1 and 2, the total initial momentum, the total final momentum, and the ratio of total final to total initial momentum.

Create a third data table to help analyze the data with columns for: run number, initial kinetic energy for carts 1 and 2, final kinetic energy for carts 1 and 2, the total initial kinetic energy, the total final kinetic energy, and the ratio of total final to total initial kinetic energy.

After data collection and analysis, have the students answer the following questions:

What must the ratio of final to initial momentum be for momentum to be conserved?

What must the ratio of final to initial kinetic energy be for kinetic energy to be conserved?

Is momentum conserved in the collisions? Explain how this can be determined.

In kinetic energy conserved in the collisions? Explain how this can be determined.

The graph of the collision on a v vs t graph.

Helpful Hints

• The track must be completely horizontal to get accurate data.

Resources

Teacher Internet Sources

• An article discussing the benefits of teaching kinematics last--or "upside down"

• Teaching resources from the Institute of Physics related to their episode about conservation of momentum. Website: [3]

• comPADRE is a website of physics and astronomy education communities. This "link" gives instructions for a demonstration about elastic and inelastic collisions using "happy" and "sad" balls.

• The Physics Classroom. (2009). Momentum and Its Conservation. Retrieved September 28, 2009, from The Physics Classroom Website: "http://www.physicsclassroom.com"

• "The Physics Teacher Online" publishes papers on the teaching of physics, and on topics such as contemporary physics, applied physics, and the history of physics—all aimed at the introductory-level teacher.

The Physics Teacher Online. (2009). Retrieved September 13, 2009, from The Physics Teacher Online Web site: http://scitation.aip.org/tpt/

• "American Association of Physics Teachers" is the professional organization for physics teachers and scientists who are dedicated to developing an understanding and appreciation for physics through the art and science of teaching.

The American Association of Physics Teachers. (2009). Retrieved September 13, 2009, from The American Association of Physics Teachers Website: http://www.aapt.org/aboutaapt/

• A clip on YouTube Using a bowling ball and a different person's face (rather than your own) to demonstrate conservation of energy.