Conservation of Linear Momentum
Inquiry Lab Kit for AP® Physics 1
Materials Included In Kit
Spheres, steel, ¾" dia., 30 Spheres, steel, ⅝" dia., 6
Rubber bands, 6 Vtracks with wooden support feet, 6
Additional Materials Required
Air source Air track Balance, 0.01g precision
Gliders, variable masses Photogate timers
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 50minute class periods. It is important to allow time between the Introductory Activity and the GuidedInquiry Activity for students to discuss and design the guidedinquiry procedures. Also, all studentdesigned 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 guidedinquiry 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 guidedinquiry 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 postcollision 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 guidedinquiry 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 guidedinquiry 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 onedimensional situations and only qualitatively in twodimensional 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.
Correlation to Next Generation Science Standards (NGSS)^{†}
Science & Engineering Practices
Developing and using models Planning and carrying out investigations Analyzing and interpreting data Using mathematics and computational thinking Constructing explanations and designing solutions
Disciplinary Core Ideas
HSPS3.A: Definitions of Energy HSPS3.B: Conservation of Energy and Energy Transfer HSPS3.C: Relationship between Energy and Forces HSETS1.A: Defining and Delimiting Engineering Problems HSETS1.B: Developing Possible Solutions
Crosscutting Concepts
Patterns Cause and effect Scale, proportion, and quantity Energy and matter
Performance Expectations
HSPS31. 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. HSPS32. Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motion of particles (objects) and energy associated with the relative position of particles (objects). HSPS33. Design, build, and refine a device that works within given constraints to convert one form of energy into another form of energy. HSETS13. Evaluate a solution to a complex realworld problem based on prioritized criteria and tradeoffs that account for a range of constraints, including cost, safety, reliability, and aesthetics, as well as possible social, cultural, and environmental impacts.
Answers to Prelab Questions
 A rocket with mass m = 223 kg is fired into the sky and moves along a positive yaxis with velocity v = 122 m/s. The rocket explodes into two pieces and one piece (mass = 105 kg) continues to move along the positive yaxis with velocity v = 178 m/s. At what speed does the other piece move?
P_{i} = mv P_{f1} = m_{1}v_{1} and P_{f2} = m_{2}v_{2} P_{f} = P_{f1} + P_{f2} = m_{1}v_{1} + m_{2}v_{2} P_{i} = P_{f} mv = m_{1}v_{1} + m_{2}v_{2} (223 kg)(122 m/s) = (105 kg)(178 m/s) + (223 kg –105 kg)(v_{2}) v_{2} = 72.17 m/s
 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.
 Describe the differences between a onedimensional collision and a twodimensional collision. Give an example of each.
In a onedimensional 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 twodimensional 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 twodimensional.
 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}.
 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} m_{1}v_{1i} = m_{1}v_{1f} + m_{2}v_{2f} (0.25 kg)(66 m/s) = (0.25 kg)(0 m/s) + (0.25 kg)(v_{2f}) v_{2f} = 66 m/s
 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 precollision velocity of the small glider and the after velocity corresponds to the postcollision velocity of the large glider.
{13788_Data_Table_1}
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 precollision velocity of the large glider and the after velocity corresponds to the postcollision velocity of the small glider.
{13788_Data_Table_2}
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 precollision velocity of the small glider and the after velocity corresponds to the postcollision velocity of the small/large glider combination.
{13788_Data_Table_3}
Answers to Questions
GuidedInquiry Discussion Questions
 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 precollision (before) and postcollision (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.
 Set up the metal Vtrack 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.
 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.
 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.
 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 twosphere system, which stops.
 Predict the result of rolling two spheres, at a higher speed than in step 5, into a stationary threesphere system.
Two spheres are knocked away from the stationary series with the same velocity as the incident twosphere system, which stops.
 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, m_{1}v_{1i} = m_{1}v_{1f} + m_{2}v_{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 postcollision speeds do in fact equal the precollision 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 postcollision speed equal to the speed of the incident spheres. In the latter case, four spheres should move at a postcollision speed equal to the speed of the incident sphere(s).
 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 twosphere system would then be equal to ½(2m)(v/2)^{2}, or ¼mv^{2}. The kinetic energy of the twosphere system is onefourth 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 precollision velocity.
 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 postcollision, 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
 Based on the data collected in the guidedinquiry 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 postcollision 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 guidedinquiry 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.
 What factors cause discrepancies between a system’s calculated precollision momentum and postcollision momentum?
Friction between the gliders and air track and between the spheres and Vtrack, 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.
 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.
 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: AlgebraBased and Physics 2: AlgebraBased Curriculum Framework; The College Board: New York, NY, 2014.
