Kinematics, Impulse, and Human Running

Kinematics, Impulse, and Human Running Purpose This lesson explores how kinematics and impulse can be used to analyze human running performance. Students will explore how scientists determined the physical factors that allow elite runners to travel at speeds far beyond the average jogger. Audience This lesson was designed to be used in an introductory high school physics class. Lesson Objectives Upon completion of this lesson, students will be able to: ஃ describe the relationship between impulse and momentum. ஃ apply impulse-momentum theorem to explain the relationship between the force a runner applies to the ground, the time a runner is in contact with the ground, and a runner’s change in momentum. Key Words aerial phase, contact phase, momentum, impulse, force Big Question This lesson plan addresses the Big Question “W​ hat does it mean to observe?” Standard Alignments ஃ Science and Engineering Practices ஃ SP 4.​ Analyzing and interpreting data ஃ SP 5.​ Using mathematics and computational thinking ஃ MA Science and Technology/Engineering Standards (2016) ஃ HS-PS2-10(MA)​. Use algebraic expressions and Newton’s laws of motion to predict changes to velocity and acceleration for an object moving in one dimension in various situations. ஃ HS-PS2-3.​ Apply scientific principles of motion and momentum to design, evaluate, and refine a device that minimizes the force on a macroscopic object during a collision. ஃ NGSS Standards (2013) 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.

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ஃ Common Core Math/Language Arts Standards CCSS.ELA-LITERACY.RST.11-12.7.​ Integrate and evaluate multiple sources of information presented in diverse formats and media (e.g., quantitative data, video, multimedia) in order to address a question or solve a problem. Misconceptions Addressed ஃ This lesson addresses misconceptions about forces and impulse, including: ஃ The object with a larger mass exerts more force in an action-reaction pair. (Question 3) ஃ A change in momentum only happens due to a change in the force of the impact, not the change in time. (Question 9) ஃ Further information about student misconceptions on this topic can be found ​here ​ and ​here​. Primary Sources ஃ Bite “​ ​What Limits How Fast You Can Run?”​ based on​: Weyand, Peter G., Deborah B. Sternlight, Matthew J. Bellizzi, and Seth Wright. 2000. “​Faster top running speeds are achieved with greater ground forces not more rapid leg movements​.” ​Journal of Applied Physiology​ 89(5): 1991–1999. ஃ Misconceptions ஃ Hestenes, David, Malcolm Wells, and Gregg Swackhamer. 1992. "F​ orce concept inventory.​ " ​The Physics Teacher 3​ 0(3): 141–58. doi:10.1119/1.2343497. ஃ Singh, Chandralekha, and David Rosengrant. 2003. "​Multiple-choice test of energy and momentum concepts.​ " ​American Journal of Physics ​ 71(6): 607–17. doi:10.1119/1.1571832. Materials Copies of the student handout and Science Bite for each student Time This lesson should take approximately one or two 50-minute class periods. Student Prior Knowledge Prior to conducting this lesson, students should be able to provide conceptual definitions of linear motion, perform quantitative analysis of kinematics, momentum, and impulse, relate forces to velocity and acceleration, and describe impulse as related to force and time, and to momentum. Instructions and Teacher Tips ஃ General Procedure ஃ Ask students students to brainstorm where the act of running may occur in their lives. As

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students offer suggestions record them on the board. Have students work together to rank which activities require the fastest running motion. For example, a list may include, being late to class, playing sports, running outside, moving through an airport, etc. ஃ Have students read the Introduction and answer Questions 1–6 in groups. ஃ Review the answers to the questions with the class. In your discussion, make sure to review the information in the paragraph after Question 6. Students may accidentally skip this part, but it is important in understanding why a runner traveling at a constant velocity experiences an impulse with each step. ஃ As you discuss human running, you may want to call up a volunteer to model the different phases of running and what differences in stride frequency, stride length, contact time, aerial time, and maximum force applied to the ground would look like. ஃ You could also show the video “Slow motion running - side view” by EMU Running Science Laboratory, posted 2015. h​ ttps://youtu.be/Jd8Jijb7jZY​ in order to aid your discussion. ஃ Have students read the Bite and answer Questions 7–11. ஃ Review the answers with the class. ஃ Tips, Extensions, and Variations ஃ You could have students collect their own data and compare it to the researchers’ plots. In order to do this groups of 2–4 students must make a video recording of themselves running on a long piece of paper with wetted shoes or bare feet. They will also need to measure the total distance they ran and the time it took them to run that distance (the time could be determined from the video or measured separately using a stopwatch). ஃ To determine stride length, students can measure the distance between one left (or right) footfall and the next left (or right) footfall. It is recommended that they measure multiple stride lengths and then average them. ஃ Students will then need to use video analysis software such as that available on Vernier’s LoggerPro or the free software Tracker to determine aerial time and contact time. They can do this by analyzing the time of the frames when their foot is in contact with the ground and those when their feet are not in contact with the ground. In order to get an accurate contact time, students must multiple the time they determine by 0.85. This factor accounts for the fact that the video doesn’t necessarily record the precise moments where a runner’s foot starts and finishes force delivery to the ground (due to shoe cushioning, etc). Scientists studying runners have simultaneously a) used force plates to measure exactly when forces are being applied to the ground by the foot and b) used videos like in this exercise to identify when the foot strikes and lifts off the ground. When they compared the estimates of contact time from video (b) to real contact time measured by force plates (a), they found that we consistently overestimate contact time from video. Real contact time was ~85% of the values predicted from video analysis, and therefore, students should adjust their video-derived contact time accordingly.

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ஃ Students can then use the equation below to calculate maximum force scaled by body weight. This “scaling” means that the value students calculate here equals the force their runner created in newtons divided by their body weight in newtons. Therefore the number they calculate here has no units. This equation was determined by scientists who have studied the relationship between maximum force applied and running speed. In order to determine the maximum force, students first must determine running speed by dividing their total distance by their total time. Maximum force = [1.26 + (0.101 × running speed)] ஃ The class can then plot all of the data in order to make graphs like those in ​Figure 2 in the Student Handout. Each group will contribute one point on each graph. Hopefully the class will see similar trends to the researchers. If not, a discuss possible errors with the students. Example errors include: students not running normally, because of the odd situation, difficulty identifying the first and last frames of contact phase, difficulty measuring the stride length due to the runner not running in a straight line, etc. ஃ Newton’s laws of motion can help explain the concept of walking around in everyday life. The various different surface contacts between feet/shoes and ground affects the frictional coefficient, which in turn works against motion. The current field of sports science engages questions like these to study human performance analysis, making runners faster. You may consider the following prompts in order to push student thinking further: ஃ Walking on ice versus pavement, then running on ice versus pavement. Why does motion on smoother surfaces lead to slipping? ஃ Various styles of athletic footwear such as ballet slippers, cleats, bowling shoes, track shoes, spikes, flip flops, etc. Why do different sports use different shoes? How does friction impact athlete performance? Direct student conversation to blend from shoe style to corresponding leg motion needed to perform activities for respective shoes. Note the comparison of leg mechanics versus desired motion outcome. ஃ If students are interested, the impulse-momentum theorem can be derived quickly from Newton’s second law of motion and the equation for acceleration: Δv Δv F = ma and a = Δt , substitute Δt for acceleration Δv F = m Δt , rearrange F Δt = mΔv Big Question Discussion This lesson plan addresses the Big Question​ “What does it mean to observe?” ​ In particular, why do scientists collect data on things that don’t seem to actually matter? If you choose to delve into the Big Question, consider the following ideas:

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ஃ Make a class list of times that students observed something they didn’t think was important, but later was key. If students can’t think of examples in their own lives, discuss examples of television or movie detective shows. Connect their examples to the experiment described in the Bite. ஃ Make a class list of times that students wished they had made note of an observation. This can be as simple as wishing you had written down which area of a parking garage you parked in or where you put your keys. This can also be linked to experiments that have been performed in class. Connect their examples to the experiment described in the Bite. Background Information and Research Details ஃ In their introduction to the research paper, the authors mention anatomical reasons for increased speed in humans. These include thin legs that have quick muscle fibers which allows legs to change positions quicker. Additionally, longer legs allow for increased stride length due to creating more propulsion forward. This article attempted to explain why these features help increase human speed. ஃ In this study, authors performed their tests on 33 subjects with a variety of sprinting abilities. Additionally, they tested the effects of having a inclined and declined treadmill on five different subjects. The top speeds for the inclined vs. declined treadmills were significantly different, with the declined top speed being 1.4 times greater than the inclined speeds. The authors determined that the greater speeds achieved at a decline were due to forces applied to the ground that depended on the subjects’ masses. This helped support the results that greater forces applied to the ground, not faster leg repositioning in the air, caused greater top speed in humans. This also helps explain why longer contact time can lead to greater speed - if the force you apply to the ground is exerted on the ground for a longer period of time, your change in momentum and therefore your velocity change will be greater. Answers 1. Describe ​and e​ xplain the three ways a runner can increase their speed. Method 1: The runner could increase their stride frequency. This means increasing how fast they take steps. Method 2: The runner could increase their contact time, or how long their feet are on the ground. This means increasing how long they apply force to the ground. Method 3: The runner could increase the amount of force they use to push off the ground. 2. In the space provided, define the two phases of running in 1–2 sentences. Aerial: Aerial phase is the part of running where both of the runner’s feet are off of the ground after one leg pushes off and before the other leg lands on the ground.

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Contact: Contact phase is the part of running where one of the runner’s feet is on the ground. 3. During which phase is the runner exerting a force to the ground? What does the ground do in return? Explain. (​Hint:​ Recall Newton’s laws of motion.) Newton’s third law says every action has an equal and opposite reaction, which means that the force can only be applied when there are two objects in contact. So, the runner can only apply force to the ground in contact phase because the runner’s feet are on the ground then. The ground is pushing back on the runner with an equal and opposite force. As the ground exerts a force on the runner, it also creates an impulse. Recall the Impulse equation:

Impulse = F orce × change in time = F Δt

Δt = tfinal − tinitial 4. What information do you need to calculate your runner’s impulse during each step? To calculate impulse, I need to measure the force the runner pushes off the ground with and the time the runner’s feet are on the ground. Researchers have examined how stride frequency, aerial time, contact time, and the maximum force that someone pushes off the ground with impacts how fast they run. Take a look at the graphs below and answer Question 5.

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5. Let’s explore the relationship between each variable and running speed. a. Fill in the table to describe what happens to running speed as each of the variables listed increases. Stride frequency is filled in for you as an example.

Stride Maximum force applied to Frequency Stride length Contact time Aerial time running surface Graph a Graph b Graph c Graph d Graph e

As stride frequency As stride length As contact There is no As maximum force increases, the increases, running time consistent increases, running speed running speed speed increases. increases, relationship increases. increases. running speed between aerial decreases. time and

running speed.

b. Based on your answers in the table above, which factor or factors do you think allow runners to run faster? A runner’s stride frequency, stride length, contact time, and maximum force seem to allow runners to run faster. c. Which factor or factors do not appear to be related to running speed? Aerial time does not seem to be related to running speed. d. Based on your answers to part a and Question 4, as a runner increases their speed, does the impulse they experience when they are touching the ground change? Justify your response. Impulse might be increasing a little bit as they run faster because the force is increasing a lot but the contact time decreases a little. So, impulse can’t change a lot because of contact time’s decrease, but since the force change is bigger, impulse might be getting larger too.

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6. Based on these equations and your answer to question 5d, how do you think a runner’s momentum and velocity are changing during contact time? Momentum: There is an impulse during contact time, so the runner’s momentum must be changing. If a runner is applying a force in the direction they are already moving, the momentum must be increasing. Velocity (Hint: Think about what Newton’s laws say about force and velocity. What does applying a force mean for velocity?): The runner’s mass isn’t changing, but momentum is increasing a little bit, so the velocity must be increasing. Additionally, Newton’s laws say that velocity is constant unless a force is applied. Because of this, velocity has to be changing during contact time. Again, if the force is in the same direction the runner is already moving, the runner’s velocity must be increasing. Think about a runner jogging in a straight line in park. Over the course of five or ten minutes, the runner’s mass and velocity are constant. Based on those conditions, you might expect the runner’s change in momentum and impulse to be zero. But, this is real life, not idealized conditions like you often think about in physics class! In real life, energy is lost to heat and friction every time the runner’s foot hits the ground. So, to keep running at the same velocity, the runner needs to balance that loss by adding momentum every step. This is why you have found non-zero force, impulse, and momentum in the questions above. Let’s see if your conclusions match that of the researchers and what makes an elite runner run so much faster than the rest of us.

What Limits How Fast You Can Run? 7. Did your conclusions above match that of the researchers? Explain your answer. Our conclusions are the same as the other researchers because we both found that increasing force and frequency increases running speed but that contact time decreases with running speed. 8. What did the researchers conclude was the most important factor in allowing elite runners to move so fast? Does that agree with the graphs you examined in Question 5? Explain your answer. The researchers concluded that the force a runner applies to the ground is the most important factor in allowing elite runners to go so fast. This does agree with the graphs in Question 5. Those graphs show that with a small change in the maximum force applied, runners have a large change in speed. Runners that experience knee pain when running are often found to overstride, meaning that they take steps that are too big. To reduce their pain, overstriding runners have to learn to take more steps while still running at the same speed. In other words, they need to increase their stride frequency.

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9. Considering your answers to Question 5a, let’s determine whether increasing step frequency is good or bad advice. For each part of this question, assume that the total impulse per stride does not change. a. If you take more steps in that given time window, do you end up spending a greater, the same, or a smaller percentage of that window in contact phase? If I’m taking more steps, then my feet are on the ground more, so I must spend a greater percentage in contact phase. b. Based on your answer to Question 9a, is the runner applying more, the same, or less force to the ground in each step? If impulse is constant and contact time increases, force must decrease in order for impulse to stay constant. c. Based on your answer to Question 9b, is it good advice to tell overstriding runners to increase their stride frequency while running at the same speed to reduce their pain? It is good advice as increasing stride frequency increases contact time which causes lower forces that might not hurt the runner as much. 10. Describe the physical mechanisms that are being changed when you run faster. You may want to consider what your body is doing when you change stride frequency/aerial time, contact time, or maximum force. Stride frequency means how fast I am taking steps, so increasing frequency means I’m taking steps faster. Force is about how hard I push off of the ground, so increasing force means I am pushing harder. Contact time and aerial time tell me how much time my feet are on the ground or in the air. My feet are on the ground for less time as I go faster, but I spend more time in the air. 11. Connect to the Big Question The scientists may have figured out that aerial time did not have ​ a large impact on running speed pretty early in their study, but they they kept recording that data anyway. Why was recording this data important even though it turned out not to affect the variable they were interested in? Why do you think scientists record all of the details of an experiment they are conducting? Have you made any observations that at the time didn’t seem important, but you used later on? Has there ever been a time where you wished you had recorded something that you originally didn’t think was important? Explain your answer. Sample answer: I​ t was important for scientists to record the data about aerial time to ensure that it did not in fact affect running speed. It could be possible that their initial measurements were an anomaly or that aerial time had some other relationship to the other variables that wasn’t immediately obvious at the beginning of the experiment. Only by looking at the experiment as a whole were they able to conclude that aerial time did not have an effect. In general, scientists record all of the details of an experiment in order to ensure that their experiment is controlled and so that they are able to discuss trends among all of the variables that could affect their results. I once wished I had recorded the mass of a metal in a specific heat lab. I had to redo my experiment, because I had missed this crucial piece of information.

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