B3 Momentum Key Question: How Well Is Momentum Conserved in Collisions?

INVESTIGATION B3 Collaborative Learning This investigation is Exploros-enabled for tablets. See page xiii for details.

B3 Momentum Key Question: How well is momentum conserved in collisions?

The law of conservation of momentum is the second of Materials for each group the great conservation laws in physics, after the law of yy Colliding pendulum kit conservation of energy. In this investigation, students observe elastic collisions between balls of the same and yy Physics stand* differing masses. The speed of each ball before and after yy Two photogates* each collision is determined, and the total momentum yy DataCollector* before and after each collision is calculated. Students yy Calculator* compare the total momentum before and after each collision to determine how well momentum is conserved. yy Balance or digital scale, accurate to 1 gram (for the class)* Learning Goals *provided by the teacher ✔✔Perform elastic collisions between balls of various masses. Online Resources Available at curiosityplace.com ✔✔Calculate the velocity and momentum of the balls before and after each collision. yy Equipment Video: Colliding Pendulum ✔✔Determine whether momentum is conserved in yy Skill and Practice Sheets each collision. yy Whiteboard Resources yy Animation: Changes in Momentum GETTING STARTED yy Science Content Video: Newton’s Third Law yy Student Reading: Newton’s Third Law and Momentum Time 100 minutes Setup and Materials 1. Make copies of investigation sheets for students. 2. Watch the equipment video. 3. Review all safety procedures with students.

NGSS Connection This investigation builds conceptual understanding and skills for the following performance expectation. HS-PS2-2. Use mathematical representations to support the claim that the total momentum of a system of objects is conserved when there is no net force on the system.

Science and Engineering Practices Disciplinary Core Ideas Crosscutting Concepts

Using Mathematics and Computational Thinking PS2.A: Forces and Motion Systems and System Models

Colliding Pendulum 41 Momentum

Vocabulary on each object’s mass. The higher an object’s mass, the collision – occurs when two or more objects hit more force it takes to deflect its motion. each other There are two types of collisions: elastic and inelastic. elastic collision – a collision in which the total kinetic When an elastic collision occurs, objects bounce off energy remains the same before and after the collision each other with no loss in the total kinetic energy of the inelastic collision – a collision in which the total kinetic system. The total kinetic energy before the collision is the energy after the collision is less than it was before the same as the total kinetic energy after the collision. The collision, and which usually involves objects sticking collision between billiard balls is very close to a perfectly- together or changing shape elastic collision. law of conservation of momentum – states that in the In an inelastic collision, objects change shape or absence of external forces, the total momentum of a stick together, and the total kinetic energy of the system remains constant system decreases. The energy is not destroyed, but it is momentum – the mass of an object multiplied by transformed into forms other than kinetic energy, such as its velocity a permanent change in shape, or sound, or heat. An egg Newton’s second law – states that acceleration is force hitting the floor is one example of an inelastic collision; divided by mass two vehicles colliding is another. In both cases, some of Newton’s third law – states that for every action force, the kinetic energy in the system permanently changes an there is a reaction force equal in strength and opposite object’s shape. in direction Momentum is a property of moving matter that depends on both mass and velocity. Momentum BACKGROUND describes the tendency of objects to keep going in the same direction with the same speed. One way to look at force is that force is the action that changes momentum. A collision occurs when two or more objects hit each Conversely, any change in momentum must create force. other, and the objects exert forces on each other. Newton’s third law tells us that any time two objects hit Momentum is the product of an object’s mass and each other, the forces exerted by the objects are equal velocity. The greater an object’s momentum, the harder in magnitude and opposite in direction. However, the it is to stop. A train car moving at even a very slow speed effect of the collision on each object can differ. During is difficult to stop because its momentum is large due to a collision, momentum and energy are transferred from its mass. one object to another. The law of conservation of momentum says the total Newton’s second law explains why colliding objects momentum in a system of interacting objects cannot react differently. The second law states that an object’s change as long as all forces act only between the acceleration is directly objects in the system. If interacting objects in a system proportional to the are not acted on by outside forces, the total amount of force exerted on the momentum in the system cannot change. If one object object and inversely gains momentum, the other loses the same amount, proportional to the leaving the total unchanged. object’s mass. The force Conservation of momentum can be used to determine felt by two colliding an unknown velocity or mass if all of the other masses objects is the same, but and velocities in the collision are known. It is important to the resulting acceleration include the direction of the velocity (positive or negative) and velocity depend because velocity and momentum are vector quantities.

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5E LESSON PLAN

Engage Newton’s third law tells us that when two objects collide, they exert equal and opposite forces on each other. However, the effect of the force is not always the same. Demonstrate by rolling two balls of different mass Science Content Video Animation toward each other so they collide. Use two balls with a Newton’s Third Law Changes in Momentum significant difference in mass, such as a tennis ball and a baseball, or a ping pong ball and a golf ball. The force on each during the collision is the same, but they do not Elaborate have the same change in motion after the collision. Automakers use When studying motion related to collisions, we crash test dummies can predict how two colliding objects might move to study the effects using momentum and Newton’s third law of motion. of collisions on Momentum is the mass of an object multiplied by its passengers. Crash velocity. Because of this, you could also call it “mass test dummies contain in motion.” electronic sensors to measure the forces and accelerations Explore exerted at various Have students complete Investigation B3, Momentum. places on the body. The dummies are expensive, costing Students observe elastic collisions between balls of the more than $100,000 each, but they are also sturdy and same and differing masses. The speed of each ball before last through years of crash testing. and after each collision is determined, and the total momentum before and after each collision is calculated. Results of these tests have been used to make changes Students compare the total momentum before and in automobile design. The use of seat belts and airbags after each collision to determine how well momentum reduces the force on passengers by slowing down the is conserved. transfer of momentum, making today’s cars much safer than their predecessors.

Explain Consider having students study the momentum, force, Revisit the Key Question to give students an opportunity and energy changes that are inflicted on a crash test to reflect on their learning experience and verbalize dummy and how those forces can be mitigated with understandings about the science concepts explored in safety devices in an automobile. the investigation. Curiosityplace.com resources, including student readings, videos, animations, and whiteboard Evaluate resources, as well as readings from your current science yy During the investigation, use the checkpoint textbook, are other tools to facilitate student questions as opportunities for ongoing assessment. communication about new ideas. yy After completing the investigation, have students answer the assessment questions on the Evaluate student sheet to check understanding of the concepts presented.

Colliding Pendulum 43 Momentum

Explore INVESTIGATION B3 Guiding the INVESTIGATION Name ______Date ______ Setting up the experiment B3 Momentum Materials: ✔ Colliding pendulum kit If you wish to complete the investigation in a How well is momentum conserved in collisions? ✔ Physics stand shorter period of time, you can partially set up the This investigation is about momentum, a property of moving matter. An object’s ✔ Two photogates momentum depends on its velocity and mass. When a collision occurs between ✔ DataCollector equipment by attaching a hanger and arc to each two objects, momentum is transferred from one to the other. If no outside forces (such as friction) are present, the total momentum of the objects before the ✔ Calculator physics stand ahead of time. Then, during class, collision is the same as the total after the collision. ✔ Balance or digital scale, accurate to 1 gram In this investigation, you will observe collisions between objects of varying students will only have to attach and align the masses. You will calculate the momentum before and after each collision to determine how well the collisions follow the law of conservation of momentum. projectile and target balls to the correct height.

 Setting up the experiment The steel balls connect to the hanger by threading The colliding pendulum apparatus allows you to observe collisions between two balls of the same or different mass. The balls are made the strings through the holes in the two posts. You of hardened steel and the collisions are almost perfectly elastic. (In an inelastic collision, the colliding objects stick together or change shape as a can see the holes if you unscrew the post a few result of the impact.) millimeters. The posts do not unscrew completely, 1. Attach the arc at the lowest possible point in the stand. Secure it with a threaded knob. Attach the pendulum hanger, leaving nine so do not try to remove them. The posts that hold uncovered holes between the arc and the hanger. the string in place only need gentle tightening; 2. Loosen the hanger’s left post and insert the string from one of the medium-sized balls. Gently tighten the post to secure the string so over‑tightening may damage the strings. the ball hangs a little below the bottom of the arc.

3. Stand back and look at the string relative to the alignment mark on the arc. The string should be in line with the mark. Adjust the To correctly align the lengths of the pendulum leveling feet on the base of the stand if necessary. strings, first align the projectile ball by putting the 4. Loosen the post and readjust the length of the string until the bottom of the ball is 2 millimeters above the arc. Attach the other medium-sized ball to the right-hand post the starting block in the 30-degree notch of the arc. Fit same way. The two balls should be at the same level relative to the arc. the projectile ball against the hole on the front of the 5. Place two photogates in the notches at the bottom of the arc. Set the Data Collector to CPO timer mode, and select the interval function by tapping the “I” icon. Attach the right photogate to input A and the left starting block. Adjust the pendulum string so the ball photogate to input B. fits exactly against the hole. Next, let the projectile

Copyright © CPO Science B3 Momentum ball hang at its lowest point. Adjust the length of the Can be duplicated for classroom use 1 of 6 Colliding Pendulum target ball string so its center aligns with the center of the projectile ball.

If necessary, adjust the leveling feet of the physics stand to align the balls in the center of the photogates. It is important that the balls pass directly STEM CONNECTION through the center of the photogates to ensure accurate velocity readings. The physics stand pole Accident reconstruction Police forensics does not need to be completely vertical. It may lean specialists use conservation of momentum forward, but shouldn’t lean to the side. and other physics knowledge to analyze traffic accidents. Skid marks, debris such as broken glass, and other clues allow investigators to m reconstruct the events of an accident scene 2 with surprising accuracy. v Skid marks are used to 2 determine the directions of the vehicles before and after m1 v the crash. Skid marks can 1 also be used to estimate velocities. With information on friction, skid distances, and directions, forensic specialists use momentum conservation to determine the vehicles’ velocities before and after the crash.

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Explore INVESTIGATION B3 Explore INVESTIGATION B3

6. Slide the starting block into the 30-degree notch on the arc. To release the ball, grip it between two 5. How does the momentum of the target ball after the collision compare to the momentum of the projectile fingers and hold it against the hole in the front side of the starting block. The hole positions the projectile ball before the collision? Calculate the percent difference between the two momentums. ball so it will hit the center of the target ball. Release the ball by opening your fingers evenly, allowing The momentum of the target ball after the collision is very close to the ball to drop straight out of the hole. the momentum of the projectile ball before the collision. It is only 7. It can be tricky to get the balls to collide head-on. You can tell when this has occurred because the target ball moves straight along the same line that the projectile ball followed. Practice dropping the projectile very slightly less, with a difference of 3.6%. ball until you can get a head-on collision most of the time.  Larger projectile colliding with smaller target  Collisions with equal masses Remove the projectile ball and replace it with the largest ball. Align The centers When recording data during this experiment, only include measurements for head-on collisions. Repeat the the two balls so their centers are at the same height. of the balls trial and do not record the data if one or both of the balls moves to the side after the collision. should be on 1. Make a hypothesis to predict what you think will happen when the same line 1. With the target ball at rest, drop the projectile ball. Catch the target ball when it flies up. Record the times the two balls collide. for trial 1 in Table 1. Repeat for a total of 3 trials and find the averages. The projectile will slow down but keep Table 1: Time data for equal mass balls Target moving, and the target will move quickly. Projectile Trial 1 Trial 2 Trial 3 Average Projectile time (s) before collision 0.0218 0.0217 0.0217 0.0217 2. Drop the projectile ball and observe the collision. Describe what happens to each ball. Was your hypothesis correct? Target time (s) after collision 0.0225 0.0225 0.0226 0.0225 The projectile ball slows down but keeps moving after the collision.

2. Measure the mass of each ball and record it in Table 2. The target ball moves away from the target ball quickly. Table 2: Velocity and momentum for equal mass balls 3. Perform another collision and catch the balls right after the collision, before they fall back through the photogates. In this collision, both balls pass through photogate B. To get the time for the target ball after Mass (g) Diameter (cm) Velocity (cm/s) Momentum (g∙cm/s) the collision, you must select memory by clicking the “m” at the bottom of the DataCollector screen when the B light is on. Record the times for trial 1 in Table 3. Repeat for a total of 3 trials and find the Projectile before collision 2.54 66.6 116.9 7,784 average times. Target after collision 66.6 2.54 112.7 7,507 Table 3: Time data for larger projectile/smaller target collision

3. Calculate the velocity of each ball. The distance each ball moves while it blocks the photogate Trial 1 Trial 2 Trial 3 Average beam is equal to the ball’s diameter. For the time, use the “average” values in Table 1. Velocity is direction-dependent. Decide in your group which direction is positive and which is negative. The Projectile time (s) before collision 0.0282 0.0282 0.0282 0.0282 projectile’s initial direction of travel is usually considered positive. (photogate A) Answers are shown in Table 2. Projectile time (s) after collision (photogate B) 0.1096 0.1095 0.01125 0.1105 4. Calculate the momentum of each ball by multiplying its mass by its velocity. Target time (s) after collision Answers are shown in Table 2. (photogate B - memory) 0.0173 0.0172 0.0170 0.0172

Guiding the INVESTIGATION

 Collisions with equal masses To make a collision, pull the target ball up the arc and hold it against the starting block between your thumb and first finger, with your hand over the top of the ball and starting block. Release the ball carefully by opening your fingers. You want the ball to swing straight down the arc with no sideways motion. A sideways motion will cause the target ball to move off at an angle during the collision. You want to make head-on collisions where the target ball moves in the same direction as the projectile ball was originally traveling. It takes some practice to get the balls to Encourage students to be patient and give each group collide perfectly head on. member a chance to try creating collisions. They may have to perform each collision several times to get valid velocity data. They should only record data for perfect head-on collisions.

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Explore INVESTIGATION B3 Explore INVESTIGATION B3

4. Record the masses, velocities, and momentums in Table 4. 2. Drop the projectile ball and observe the collision. Describe what happens to each ball. Was your Table 4: Velocity and momentum for larger projectile/small target collision hypothesis correct? The projectile ball bounced back, and the target ball moved slowly Mass (g) Diameter (cm) Velocity (cm/s) Momentum (g∙cm/s) away from the projectile ball. Projectile before collision 129.4 3.18 112.8 14,592

Projectile after collision 129.4 3.18 28.8 3,723 3. To measure the times, catch the balls right after a collision, before they fall back through the photogates. Allow the projectile ball to pass through photogate A twice. To get the time before the collision, you must select memory by clicking the “m” icon at the bottom of the DataCollector screen. Record data for three Target after collision 2.54 66.6 148.0 9,854 trials in Table 5. Total after collision 13,577 Table 5: Time data for smaller projectile/larger target collision

Trial 1 Trial 2 Trial 3 Average 5. Add the momentums of the projectile and target balls to find the total momentum after the collision. How does it compare to the projectile’s momentum before the collision? Calculate the percent difference Projectile time (s) before collision between the momentum before and after the collision. (photogate A - memory) 0.0223 0.0222 0.0224 0.0223 Total before collision = 14,592 (projectile momentum) Projectile time (s) after collision 0.0910 0.0883 0.0898 0.0897 + 0 (target momentum) = 14,592 (photogate A) Percent difference = (14,592 – 13,577)/14,592 = 0.07 x 100 = 7% Target time (s) after collision (photogate B) 0.0429 0.0432 0.0424 0.0428 The total momentum after the collision is slightly less than the

momentum of the projectile ball before the collision. The a. Record the masses, velocities, and momentums in Table 6. The projectile ball reverses direction during difference is 7%. the collision, so it has a negative velocity and momentum after the collision. Table 6: Velocity and momentum for smaller projectile/larger target collision The centers  Smaller projectile colliding with larger target of the balls should be on Mass (g) Diameter (cm) Velocity (cm/s) Momentum (g∙cm/s) Switch the locations of the target and projectile balls. The projectile the same line is now the medium ball. Align the two balls so their centers are at the Projectile before collision 66.6 2.54 113.9 7,586 same height. Projectile after collision 66.6 2.54 –28.3 –1,886 1. Make a hypothesis to predict what you think will happen when the Target Projectile two balls collide. Target after collision 129.4 3.18 74.2 9,607 The projectile will bounce back, and the target will move slowly away Total after collision 7,721 from the projectile ball.

ADDRESSING MISCONCEPTIONS

Students often have difficulty with the terms elastic Be aware of students classifying all objects that and inelastic. Elastic and inelastic are conventions used bounce off each other as elastic collisions. The objects to make it easier to analyze the forces, motion, and do not have to stick together in order for the collision energy of a collision. The collisions seen in everyday to be classified as inelastic. Consider the situation life are a mix of elastic and inelastic. When two billiard where two cars crash and the cars bounce off each balls collide, it looks like they bounce without a loss other, but both have damage. This is considered an of kinetic energy. But the sound of the collision tells inelastic collision because both objects sustained you a small amount of kinetic permanent deformations. energy is being changed into sound energy. We approximate the collision of the balls as elastic because it is very close to a perfectly-elastic collision, even though a small amount of energy is lost (to sound) in the collision.

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SCIENCE AND MATH Explore INVESTIGATION B3

b. Add the momentums of the projectile and target balls to find the total momentum after the collision. How does it compare to the projectile’s momentum before the collision? Calculate the percent difference Using momentum to analyze problems can be between the momentum before and after the collision. challenging for students. Practice applying Percent difference = (7,586 – 7,721)/7,586 × 100 = 0.018% conservation of momentum in a collision with the The total momentum after the collision is very close to the following problem. projectile’s momentum before the collision. The difference is 1.8%. c. According to the law of conservation of momentum, the total momentums before and after a collision should be the same. Discuss some reasons why the momentums before and after the collisions in this Two 0.165-kg Before collision investigation may have been slightly different. billiard balls roll The momentums may have been slightly different because it was m1 m2 hard to create perfect head-on collisions. If the balls did not move toward each other straight through the photogates, the times would not be perfectly and collide head- accurate because it takes more time to move diagonally through v1 v2 on. Initially, the 10 a photogate than straight through it. The distance for which a ball ball has a velocity m = m = 0.165 kg blocked the photogate would also be inaccurate if the ball did not 1 2 go straight through or if it was slightly higher or lower than it should of 0.5 m/s. The 5 be. The distance measurements assumed that the entire diameter v1 = 0.5 m/s ball has an initial passed through the beam of the photogate. v = –0.7 m/s velocity of –0.7 2 d. Newton’s third law states that the forces on two colliding objects are equal in strength and opposite in direction. Newton’s second law explains the relationship between acceleration, force, and mass. Use m/s. The collision After collision these two laws to explain what happened during the three collisions. is elastic, and the The force felt by two colliding balls has the same strength. If the balls have equal masses, they accelerate equally. One slows down 5 ball rebounds and the other speeds up by the same amount. If the masses are with a velocity of v v unequal, the lighter ball accelerates more as a result of the force. 0.4 m/s, reversing 3 4 The lighter ball’s velocity therefore changes by a greater amount. its direction. What v = ? is the velocity of 3 v = 0.4 m/s the 10 ball after 4 the collision? Copyright © CPO Science B3 Momentum Can be duplicated for classroom use 6 of 6 Colliding Pendulum Looking for: The velocity of the 10 ball after the collision

Given: The initial mass and velocity of both balls. You are also given the velocity of the 5 ball after the collision.

Relationships: Both balls are of equal mass. Using the conservation of momentum, the sum of the momentums

of both balls before the collision (mv1 + mv2) is equal to the sum of the momentums of both balls after the

collision (mv3 + mv4). Solution:

mv1 + mv2 = mv3 + mv4

(0.165 kg)(0.5 m/s) + (0.165 kg)(−0.7 m/s) = (0.165 kg)(v3) + (0.165 kg)(0.4 m/s)

−0.033 kg i m/s = (0.165 kg)(v3) + 0.066 kg i m/s

v3 = −0.6 m/s

The 10 ball travels at –0.6 m/s, the negative value indicating its movement in the opposite direction as shown by the arrow in the diagram.

Colliding Pendulum 47 Momentum

Notes and Reflections Evaluate INVESTIGATION B3

Name ______Date ______

1. Which object has the most momentum? Circle the correct answer. a. A 5-kilogram cat running at 5 m/s c. A 50-kilogram person walking at 1 m/s b. A 1,000-kilogram car that is not moving d. A 1-kilogram bird flying at 15 m/s

2. In which scenario will the projectile ball continue moving in its original direction after the collision? Circle the correct answer. a. A large projectile ball hits a small target ball b. A small projectile ball hits a large target ball c. A medium projectile ball hits a medium target ball

3. In which scenario will the target ball’s velocity after the collision equal the projectile ball’s velocity before the collision? Circle the correct answer. a. A large projectile ball hits a small target ball b. A small projectile ball hits a large target ball c. A medium projectile ball hits a medium target ball Before collision After collision 4. A 50-gram ball moving at 6 m/s hits a 10-gram ball at rest. After the 50 grams 50 grams collision, the 10-gram ball is moving at 10 m/s. 10 grams 10 grams a. Calculate the momentum of the 50-gram ball before the collision.

50 g × 6 m/s = 300 g∙m/s stopped 10 m/s 6 m/s ?

b. Calculate the momentum of the 10-gramBefore ball collision after the collision. After collision 50 grams 50 grams 10 g × 10 m/s = 100 g∙m/s 10 grams 10 grams

c. How much momentum must the 50-gram ball have after stopped 10 m/s the collision? 6 m/s ? The total momentum must add up to 300 g∙m/s, so the 50-gram ball must have a momentum of 200 g∙m/s.

d. Calculate the velocity of the 50-gram ball after the collision. 200 g∙m/s ÷ 50 g = 4 m/s.

WRAPPING UP

Have your students reflect on what they learned from the investigation by answering the following questions: 1. What is momentum? 2. How is momentum calculated? 3. What does it mean to say momentum is conserved?