Materials Included In Kit

Additional Materials Required

Safety Precautions

The materials in this lab are considered nonhazardous. Follow all laboratory safety guidelines.

Disposal

All materials may be saved and stored for future use.

Lab Hints

• This laboratory activity can be completed in two 50-minute class periods. It is important to allow time between the Introductory Activity and the Guided-Inquiry Activity for students to discuss and design the guided-inquiry procedures. Also, all student-designed procedures must be approved for safety before students are allowed to implement them in the lab. Prelab Questions may be completed before lab begins the first day.
• This investigation has been written to be educational and economical. Should multiple air tracks be available, the Introductory Activity can be run as an introductory experiment by groups of 3–4 students. Moreover, the Introductory Activity can be converted to a guided-inquiry activity by omitting some procedural information. For example, rather than instructing students to nudge the large glider into the small glider, students can be asked to design an experiment using an air track to confirm that momentum is conserved in elastic and inelastic collisions. The level of difficulty can be increased by withholding information or decreased by providing hints.
• The investigation’s guided-inquiry portion, though qualitative, yields very good data. Students can run many trials in a short amount of time and refine their experimental designs to find incident sphere velocities that allow for easy qualitative description of post-collision velocities. Moreover, students can explore kinetic energy conservation using spheres of different mass.
• The Internet contains a number of videos in which gliders collide with each other on air tracks. Such videos may be used in the place of an air track should one not be available. As a class, examine the video carefully prior to conducting the guided-inquiry activity. Ask students what calculations are necessary to confirm linear momentum and kinetic energy conservation given velocity data.

Teacher Tips

• The fundamental realization this investigation should induce in students, via guided-inquiry and critical thinking, is that total kinetic energy is conserved in an elastic collision. For that reason a single sphere cannot knock away more than one sphere of equal mass from a stationary series, regardless of incident velocity. Such a scenario does not maintain the system’s total kinetic energy. In contrast, students should also realize that total kinetic energy is not conserved in inelastic collisions.

Further Extensions

Opportunities for Inquiry

The collisions considered in this investigation all have a moving object and a stationary object. To further explore the principle of linear momentum conservation, design an experiment in which moving gliders or spheres collide with other moving gliders or spheres. Collisions in which the objects move in opposite directions and in the same direction can be carried out.

Alignment to the Curriculum for AP^® Physics 1

Enduring Understandings and Essential Knowledge
Certain quantities are conserved, in the sense that the changes of those quantities in a given system are always equal to the transfer of that quantity to or from the system by all possible interactions with other systems. (5A)
5A2: For all systems under all circumstances, energy, charge, linear momentum, and angular momentum are conserved. For an isolated or a closed system, conserved quantities are constant. An open system is one that exchanges any conserved quantity with its surroundings.
5A4: The boundary between a system and its environment is a decision made by the person considering the situation in order to simplify or otherwise assist in analysis.

The linear momentum of a system is conserved. (5D)
5D1: In a collision between objects, linear momentum is conserved. In an elastic collision, kinetic energy is the same before and after.
5D2: In a collision between objects, linear momentum is conserved. In an inelastic collision, kinetic energy is not the same before and after the collision.

Learning Objectives
5D1.1: The student is able to make qualitative predictions about natural phenomena based on conservation of linear momentum and restoration of kinetic energy in elastic collisions.
5D1.2: The student is able to apply the principles of conservation of momentum and restoration of kinetic energy to reconcile a situation that appears to be isolated and elastic, but in which data indicate that linear momentum and kinetic energy are not the same after the interaction, by refining a scientific question to identify interactions that have not been considered. Students will be expected to solve qualitatively and/or quantitatively for one-dimensional situations and only qualitatively in two-dimensional situations.
5D1.3: The student is able to apply mathematical routines appropriately to problems involving elastic collisions in one dimension and justify the selection of those mathematical routines based on conservation of momentum and restoration of kinetic energy.
5D1.4: The student is able to design an experimental test of an application of the principle of the conservation of linear momentum, predict an outcome of the experiment using the principle, analyze data generated by that experiment whose uncertainties are expressed numerically, and evaluate the match between the prediction and the outcome.
5D1.5: The student is able to classify a given collision situation as elastic or inelastic, justify the selection of conservation of linear momentum and restoration of kinetic energy as the appropriate principles for analyzing an elastic collision, solve for missing variables, and calculate their values.
5D2.1: The student is able to qualitatively predict, in terms of linear momentum and kinetic energy, how the outcome of a collision between two objects changes depending on whether the collision is elastic or inelastic.
5D2.3: The student is able to apply the conservation of linear momentum to a closed system of objects involved in an inelastic collision to predict the change in kinetic energy.
5D2.4: The student is able to analyze data that verify conservation of momentum in collisions with and without an external friction force.
5D2.5: The student is able to classify a given collision situation as elastic or inelastic, justify the selection of conservation of linear momentum as the appropriate solution method for an inelastic collision, recognize that there is a common final velocity for the colliding objects in the totally inelastic case, solve for missing variables, and calculate their values.

Science Practices
2.2 The student can justify the selection of a mathematical routine to solve problems.
3.1 The student can pose scientific questions.
4.1 The student can justify the selection of the kind of data needed to answer a particular scientific question.
4.2 The student can design a plan for collecting data to answer a particular scientific question.
4.3 The student can collect data to answer a particular scientific question.
4.4 The student can evaluate sources of data to answer a particular scientific question.
5.1 The student can analyze data to identify patterns or relationships.
5.2 The student can refine observations and measurements based on data analysis.
5.3 The student can evaluate the evidence provided by data sets in relation to a particular scientific question.
6.1 The student can justify claims with evidence.
6.2 The student can construct explanations of phenomena based on evidence produced through scientific practices.
6.3 The student can articulate the reasons that scientific explanations and theories are refined or replaced.
6.4 The student can make claims and predictions about natural phenomena based on scientific theories and models.
7.2 The student can connect concepts in and across domain(s) to generalize or extrapolate in and/or across enduring understandings and/or big ideas.

Answers to Prelab Questions

1. A rocket with mass m = 223 kg is fired into the sky and moves along a positive y-axis with velocity v = 122 m/s. The rocket explodes into two pieces and one piece (mass = 105 kg) continues to move along the positive y-axis with velocity v = 178 m/s. At what speed does the other piece move?

   P_i = mv
   P_f1 = m₁v₁ and P_f2 = m₂v₂
   P_f = P_f1 + P_f2 = m₁v₁ + m₂v₂
   P_i = P_f
   mv = m₁v₁ + m₂v₂
   (223 kg)(122 m/s) = (105 kg)(178 m/s) + (223 kg –105 kg)(v₂)
   v₂ = 72.17 m/s

2. Describe the differences between an elastic collision and inelastic collision. Give an example of each.

   In an elastic collision, the total kinetic energy of the system is conserved. An example of an elastic collision is that between a super (bouncy) ball and a wall or floor.

   In an inelastic collision, objects collide and subsequently move as one unit. In this type of collision, total kinetic energy is not conserved. Rather, some kinetic energy is converted to other forms of energy such as thermal energy and the energy of sound. An example of an inelastic collision is that between a football and a wide receiver’s hands.

3. Describe the differences between a one-dimensional collision and a two-dimensional collision. Give an example of each.
   
   In a one-dimensional collision, the motions before and after the collision occur on a single axis. For example, the collisions between bowling balls in a lane gutter are one dimensional.
   
   In a two-dimensional collision, the motions before and after the collision are in a plane containing two axes. For example, the collisions between pucks on an air hockey table are two-dimensional.
   
   
4. Billiards, or pool, is a game governed by the principle of linear momentum conservation. In a game of billiards, a cue ball is used to strike other balls to direct them into pockets. Regardless of the cue ball’s incident speed prior to a collision, its velocity drops to zero in the moment directly after the collision. Explain.
   
   This is the case because in direct collisions, objects of equal masses exchange velocities. This is the only scenario that conserves total kinetic energy and momentum according to p_i = p_f.
   
   
5. A cue ball with mass m = 0.25 kg moves with a velocity v = 66 m/s. It strikes a stationary eight ball (mass = 0.25 kg) and projects it into a corner pocket. At what speed does the eight ball leave the collision?

   p_i = p_f
   m₁v_1i = m₁v_1f + m₂v_2f
   (0.25 kg)(66 m/s) = (0.25 kg)(0 m/s) + (0.25 kg)(v_2f)
   v_2f = 66 m/s

6. Bicyclists often ride with traffic so as to minimize the injuries resulting from a collision with a car. Why is this the case?
   
   This is the case because bicyclists and cars advancing towards each other in opposite directions have distinct velocities that essentially add when they collide, whereas a car that strikes a bicyclist moving away from it does not have as high an incident velocity relative to the bicyclist.

Sample Data

Introductory Activity

Part A. Velocity, momentum and % difference data were obtained by colliding a small (154.9 g) glider into a large (312.5 g) glider on an air track in an elastic manner. The before velocity corresponds to the pre-collision velocity of the small glider and the after velocity corresponds to the post-collision velocity of the large glider.

Part B. Velocity, momentum and % difference data were obtained by colliding a large (312.5 g) glider into a small (154.9 g) glider on an air track in an elastic manner. The before velocity corresponds to the pre-collision velocity of the large glider and the after velocity corresponds to the post-collision velocity of the small glider. Part C. Velocity, momentum and % difference data were obtained by colliding a small (154.9 g) glider into a large (312.5 g) glider on an air track in an inelastic manner. Velcro bumpers were affixed to the gliders so that they would adhere to each other following contact. The before velocity corresponds to the pre-collision velocity of the small glider and the after velocity corresponds to the post-collision velocity of the small/large glider combination.

Answers to Questions

Guided-Inquiry Discussion Questions

1. Based on the data obtained in the Introductory Activity, was momentum conserved in the elastic collisions and the inelastic collisions? Calculate deviations from ideal or expected values to justify your answer.

   For each collision, a pre-collision (before) and post-collision (after) velocity and momentum were calculated. The photogates measured the amount of time that light was blocked by the flag attached to the glider. That is, they measured how long it takes the flag to pass through the photogate. Thus, by measuring the length of the flag, the velocity was calculated by dividing the length of the flag (distance) by the time reported by the photogate. For the elastic collisions, the “before” velocity corresponded to the velocity of the glider initially outside the photogates. The “after” velocity corresponded to the velocity of the glider initially between the photogates. By weighing the gliders, the “before” and “after” momentum may be calculated according to the equation p = mv. Below is a sample calculation for an elastic collision (Part A table, first row) and an inelastic collision (Part C table, first row).

   Elastic Collision
   “Before” Velocity:

   Time = 0.2507 s, flag length = 5.0 cm, or 0.050 m
   Velocity = distance/time = 0.050 m/0.2507 s = 0.199 m/s
   Momentum, p = mv p = (glider mass)(glider velocity) p = (0.1549 kg)(0.199 m/s) = 0.031 kg x m/s

   “After” Velocity Calculation:

   Time = 1.031 s, flag length = 10.0 cm, or 0.100 m
   Velocity = distance/time = 0.100/1.031 s = 0.0970 m/s
   Momentum, p = mv p = (glider mass)(glider velocity) p = (0.3125 kg)(0.0970 m/s) = 0.030 kg x m/s

   Inelastic Collision

   “Before” Velocity:

   Time = 0.1923 s, flag length = 5.0 cm, or 0.050 m
   Velocity = distance/time = 0.050 m/0.1923 s = 0.260 m /s
   Momentum, p = mv p = (glider mass)(glider velocity) p = (0.1549 kg)(0.260 m /s) = 0.040 kg x m/s

   “After” Velocity Calculation:

   Time = 0.5634 s, flag length = 5.0 cm, or 0.050 m
   Velocity = distance/time = 0.050 m/0.5634 s = 0.089 m/s
   Momentum, p = mv p = (glider mass)(glider velocity) p = (0.1549 kg + 0.3125 kg)(0.089 m/s) = 0.041 kg x m/s

   “% Differences” were calculated by dividing the absolute value of the difference between “before” and “after” values by the smaller value, and multiplying by 100. Alternatively, the absolute value of the difference between the “before” and “after” values may be divided by their average. The % difference value is a measure of the degree to which linear momentum is conserved. Small % difference values indicate linear momentum is conserved. All of the collisions carried out in the introductory activity have % difference values below 10, indicating momentum was conserved.

2. Set up the metal V-track apparatus by sliding wooden support feet onto the ends of the metal track and wrapping a rubber band around one of the wooden feet as shown in Figure 2. The rubber band will act as a stopper for the spheres.
3. Predict the result of rolling one sphere into four stationary spheres. How many spheres will be knocked away and at what relative speed to the incident (pushed) sphere? Did the incident sphere stop, keep moving, or recoil?

   One sphere is knocked away from the stationary series. It leaves the stationary series with the same velocity as the incident sphere. The incident bearing stops.

4. Predict the result of rolling one sphere, at a higher speed than in step 3, into four stationary spheres.

   Again, one sphere is knocked away at the same velocity as the incident sphere, which stops.

5. Predict the result of rolling two nearly touching spheres into three stationary spheres.

   Two spheres are knocked away from the stationary series with the same velocity as the incident two-sphere system, which stops.

6. Predict the result of rolling two spheres, at a higher speed than in step 5, into a stationary three-sphere system.

   Two spheres are knocked away from the stationary series with the same velocity as the incident two-sphere system, which stops.

7. Explain how the data gathered in steps 3 to 6 confirm that both momentum and total kinetic energy are conserved in elastic collisions. What additional incident/stationary sphere ratios can be used to further confirm momentum and kinetic energy conservation? Explain.

   Based on the general equation governing linear momentum conservation, m₁v_1i = m₁v_1f + m₂v_2f , because the spheres have the same mass, the incident sphere(s) must slow to zero m/s immediately following collisions with the stationary sphere(s). Thus, the sphere(s) are knocked away at velocities that mathematically must equal the velocities of the incident sphere(s). Qualitative observations indicate that the post-collision speeds do in fact equal the pre-collision speeds. In addition to the combinations recommended in steps 3 to 6, three spheres may be rolled into two and four spheres may be rolled into 1. In the former case, three spheres should move at a post-collision speed equal to the speed of the incident spheres. In the latter case, four spheres should move at a post-collision speed equal to the speed of the incident sphere(s).

8. Regardless of the velocity of the incident sphere(s) in steps 3 to 6, the number of incident spheres always equals the number of spheres knocked away from the stationary series. Explain.

   Assume that two spheres are knocked away by one colliding sphere. In order to conserve momentum, the two spheres (2m) would be knocked away from the series with half the velocity of the colliding sphere [mv = 2m(½)v]. However, the kinetic energy of this two-sphere system would then be equal to ½(2m)(v/2)², or ¼mv². The kinetic energy of the two-sphere system is one-fourth the original kinetic energy and is clearly not conserved as it should be during an elastic collision. Therefore, this result is not possible. One colliding sphere will knock away only one sphere provided the masses are equal. One sphere cannot knock away two or more spheres regardless of the pre-collision velocity.

9. Design and conduct an experiment to confirm linear momentum and total kinetic energy are conserved when the number of incident sphere(s) does not equal the number of post-collision, moving spheres.

   Collide a ¾" diameter sphere into a stationary series of two ¾" diameter spheres and one ⅝" diameter sphere at the end of the series. With a high enough incident velocity a ¾" diameter sphere and ⅝" diameter sphere are knocked away from the stationary series. In order to conserve linear momentum and total kinetic energy, the smaller sphere moves at a higher velocity than the incident sphere.

Review Questions for AP^® Physics 1

1. Based on the data collected in the guided-inquiry portion of the investigation, make the case that momentum and kinetic energy were conserved for all collisions.

   In order for momentum to be conserved, the number of incident spheres and post-collision moving spheres must be the same. Moreover, the incident sphere(s) must have the same velocity as the sphere(s) knocked away from the stationary series. In all collisions carried out in the guided-inquiry portion of the investigation, this was the case. Total kinetic energy was also conserved, as the number of incident sphere(s) always equaled the number of sphere(s) knocked from the stationary series.

2. What factors cause discrepancies between a system’s calculated pre-collision momentum and post-collision momentum?

   Friction between the gliders and air track and between the spheres and V-track, will cause discrepancies between “before” and “after” values. Even on an air track, friction does exist, particularly when the gliders wobble somewhat and contact the track. Friction will convert the gliders’ kinetic energy to thermal energy. Moreover, in all collisions with the gliders, some energy is converted to elastic potential energy and the energy of sound when the bumpers are compressed. In collisions between spheres, kinetic energy is audibly converted to the energy of sound.

3. What experimental steps would need to be taken to determine whether kinetic energy is conserved when using the air track?

   In order to quantitatively determine whether kinetic energy was conserved in the air track collisions, the velocity of the incident glider would have to be measured following the collision.

4. Though total kinetic energy is not conserved in an inelastic collision, total energy must be conserved. In an inelastic collision such as the one that occurs when one football player tackles another player to the ground, describe how energy is conserved. That is, to what other forms of energy are the football players’ energy converted?

   When two football players collide, a significant amount of kinetic energy is converted to the energy of sound. Moreover, as the players hit the ground, it is compressed and some energy is converted to elastic potential energy. Moreover, players’ bodies absorb a great deal of the energy mechanically, as tackles cause muscle and bone movement. Additionally, some kinetic energy is converted to thermal energy owing to friction between the players and between the players and ground.

References

AP^® Physics 1: Algebra-Based and Physics 2: Algebra-Based Curriculum Framework; The College Board: New York, NY, 2014.

Introduction

Collisions are common and occur daily on roadways, in sports, and in crowded pedestrian walkways. When two or more objects collide, what factors determine how the objects react? In an elastic collision, the objects bounce and exchange energy. In an inelastic collision, the objects stick and energy is lost. Where does the energy go? Investigate elastic and inelastic collisions and the principles governing such collisions.

Concepts

• Kinetic energy
• Elastic collisions
• Inelastic collisions
• Momentum

Background

When an object is set in motion, the object has a property known as momentum. Momentum is calculated by multiplying the mass of the object by its velocity. A fundamental principle of physics is that the momentum of a system of objects always remains constant. This principle is known as the conservation of momentum. If objects within a system collide, the momentum of the individual objects before and after a collision may change, but the total momentum of the system will remain constant.

The general equation to show momentum before and after a collision with a stationary object that has no initial momentum is seen in Equation 1.

There are two types of collisions—elastic and inelastic. An elastic collision occurs when the colliding objects separate after the collision (e.g., the collision between a baseball and a bat or a kicker’s foot and a football). An inelastic collision occurs when the colliding objects stick together and move as one object after the collision. For example, the collision between a baseball and catcher’s mitt or a football player tackling another to the ground. In every collision, elastic or inelastic, momentum is always conserved. The main difference between the two types of collisions is that kinetic energy remains constant for elastic collisions, but not for inelastic collisions. The conservation of energy principle does not apply to an inelastic collision because much of the energy is lost as heat and sound owing to friction.

Experiment Overview

This investigation begins with a teacher demonstration and group activity to explore whether inelastic and elastic collisions on an almost frictionless surface, an air track, conserve linear momentum. Following this quantitative portion of the investigation, elastic collisions will be further modelled using a V-shaped track and ball bearings. Based on data obtained in the group activity, predictions will be made as to what happens when systems of ball bearings collide and how kinetic energy is conserved.

Materials

Prelab Questions

1. A rocket with mass m = 223 kg is fired into the sky and moves along a positive y-axis with velocity v = 122 m/s. The rocket explodes into two pieces and one piece (mass = 105 kg) continues to move along the positive y-axis with velocity v = 178 m/s. At what speed does the other piece move?
2. Describe the differences between an elastic collision and an inelastic collision. Give an example of each.
3. Describe the differences between a one-dimensional collision and a two-dimensional collision. Give an example of each.
4. Billiards, or pool, is a game governed by the principle of linear momentum conservation. In a game of billiards, a cue ball is used to strike other balls to direct them into pockets. Regardless of the cue ball’s incident speed prior to a collision, its velocity drops to zero in the moment directly after the collision. Explain.
5. A cue ball with mass m = 0.25 kg moves with a velocity v = 66 m/s. It strikes a stationary eight ball (mass = 0.25 kg) and projects it into a corner pocket. At what speed does the eight ball leave the collision?
6. Bicyclists often ride with traffic so as to minimize the injuries resulting from a collision with a car. Why is this the case?

Safety Precautions

The materials in this lab are considered nonhazardous. Follow all laboratory safety guidelines.

Procedure

Introductory Activity

1. Set up air track and gliders as shown in Figure 1. Position the photogates approximately 48 cm apart.
   {13788_Procedure_Figure_1}
2. Turn on the air flow and level the air track by adjusting the leveling foot until the gliders remain stationary.

Part A.

3. With the bumbers facing each other, place the large glider between the photogates and the small glider outside the photogates. Attach flags to block the light path in the photogates.
4. Gently nudge the small glider towards the large glider to cause an elastic collision. Be careful to avoid pushing the gliders with too much force as this can cause them to wobble on the air track and cause friction.
5. Repeat this procedure several times and make qualitative observations as to what happens following the collision.
6. Repeat steps 3 and 4 with the photogates in “Gate Mode.” The photogate reports the time required for the flag to pass completely through in seconds. Repeat several times and record data.

Part. B.

7. With the bumpers facing each other, place the small glider between the photogates and the large glider outside the photogates.
8. Gently nudge the large glider towards the small glider to cause an elastic collision.
9. Repeat this procedure several times and make qualitative observations as to what happens following the collision.
10. Repeat steps 7 and 8 with the photogates in “Gate Mode.” Repeat several times and record data.

Part C.

11. With the velcro ends facing each other, place the large glider between the photogates and the small glider outside the photogates.
12. Gently push the small glider towards the large glider to cause an inelastic collision.
13. Repeat this procedure several times and make qualitative observations as to what happens following the collision.
14. Repeat steps 11 and 12 with the photogates in “Gate Mode.” Repeat several times and record data.

Guided-Inquiry Design and Procedure

1. Based on the data obtained in the Introductory Activity, was momentum conserved in the elastic collisions and the inelastic collisions? Calculate deviations from ideal or expected values to justify your answer.
2. Set up the metal V-track apparatus by sliding wooden support feet onto the ends of the metal track and wrapping a rubber band around one of the wooden feet as shown in Figure 2. The rubber band will act as a stopper for the spheres.
   {13788_Procedure_Figure_2}
3. Predict the result of rolling one sphere into four stationary spheres. How many spheres will be knocked away and at what relative speed to the incident (pushed) sphere? Did the incident sphere stop, keep moving, or recoil?
4. Predict the result of rolling one sphere, at a higher speed than in step 3, into four stationary spheres.
5. Predict the result of rolling two nearly touching spheres into three stationary spheres.
6. Predict the result of rolling two spheres, at a higher speed than in step 5, into a stationary three-sphere system.
7. Explain how the data gathered in steps 3 to 6 confirm that both momentum and total kinetic energy are conserved in elastic collisions. What additional incident/stationary sphere ratios can be used to further confirm momentum and kinetic energy conservation? Explain.
8. Regardless of the velocity of the incident sphere(s) in steps 3 to 6, the number of incident spheres always equals the number of spheres knocked away from the stationary series. Design and conduct an experiment to confirm that linear momentum and total kinetic energy are conserved when the number of incident spheres does not equal the number of postcollision, moving spheres.
9. Explain how the data gathered in step 8 further confirms that linear momentum and total kinetic energy are conserved in elastic collisions.

Student Worksheet PDF

13788_Student1.pdf