People power: the physics of human performance Blake Laing, Ph.D. and Harold Meyer, Ph.D., MPH, Southern Adventist University OBJECTIVES 1. Calculate the mechanical power required to move a mass up an inclined plane at a constant velocity 2. Learn three measures of power expenditure of the human body: Calories/unit time The volume rate of oxygen consumption V O 2 by the body Metabolic Equivalent of Task (MET), defined as the rate of oxygen consumption at rest. 3. Learn how to estimate your own (V O 2 ) max from the Rockport 1-mile test. 4. Use a linear response approximation and proportional reasoning to estimate your own V O 2 rate from a measured heart rate and estimated (V O 2 ) max 5. Calculate the efficiency of human motion. 6. Prescribe an exercise program for two patients/clients using the ACSM walking equation. INTRODUCTION Would you be more likely to engage in cardiovascular exercise if you knew your target heartrate for optimal cardiovascular benefit? What if you could think beyond the gym and engage in any outdoor activity to meet that target HR? Do you already use, or have you considered purchasing, a fitness tracking device or smartphone app which magically generates motivational numbers and graphs? How would you like to understand these simple measurements and become a power user? You can easily calculate your current V O 2 max with a Rockport 1-mile test ( Duckwalk ). You may use a simple Google Docs spreadsheet which calculates your new target HR for each new V O 2 max measurement, no purchase necessary. How would you like to become a more discerning consumer, understanding which products can generate more meaningful data? A good app, such as Polar Flow, should let you enter a current V O 2 max to personalize your training targets. Your knowledge of the physics of work and energy, combined with some knowledge of exercise physiology, can be used to empower the stewardship of your own wellness. You can assess and track your cardiovascular fitness by estimating your V O 2 max, and you can adjust your fitness program by including elevation gain.
THEORY 1. MECHANICAL WORK Mechanical work is defined as the change in energy of a system due to external forces. The mechanical work is the product of an opposing force times the displacement along the line of force: (Eq. 1) W = F d = Fd cos θ Where θ is the angle between F and d. For example, work required to move a mass m a distance d up an inclined plane of angle φ due to the opposing force of gravity is (Eq. 2) W = mg h = mgd sin φ, The SI unit of work (or energy) is the Joule (J), where 1 J = 1 N m. 2. MECHANICAL POWER Mechanical power is defined as the rate at which mechanical work performed. The power required to perform an amount of work W in a time interval Δt is (Eq. 2) P = W t. The SI unit of power is the watt, where 1 W = 1 J/s. The mechanical power required to move a mass at a speed v up a hill with incline angle φ is (Eq. 3) P = mg Δd sin φ = mgv sin φ Δt 3. EFFICIENCY How much power is required for a body at rest to remain at rest, or for a body to remain at rest, or remain moving at a constant velocity? You know that you must expend power to walk on a level surface. You require power just to sit still and breathe! Then why would your physics professor insist that it takes no mechanical power at all for a body to move at a constant velocity with no opposing force? The reason is that the human body or a machine does not translate power to motion with 100% efficiency. (Eq. 4) what you get Efficiency = what you paid 100% Mechanical power is defined as the rate of mechanical work performed. A machine operating at 50% efficiency will exert 2 Joules of work (requiring 2 Joule of energy) for every 1 Joule of mechanical work performed, so such a machine will require 2 watts of power for every 1 watt of mechanical power. 4. UNITS OF POWER AND ENERGY IN HUMAN PERFORMANCE All athletes who finish a race have performed the same amount of work per kg bodyweight. To win the race, an athlete has to generate the most power per kg, because the work of finishing is performed in a shorter time.
Cardiovascular performance can be quantified in units of power. The SI unit of power is the watt, but power can also be measured as a rate of oxygen consumption. All aerobic organisms generate energy from food and oxygen in the citric acid cycle. The amount of energy generated in this cycle can be measured in Joules, calories, or volume of oxygen consumed. The (little-c) calorie is a unit of energy used to quantify the energy the body derives from food. The number of calories is measured by burning food in a calorimeter (1 cal = 4.19 J). Each 1 Liter O 2 consumed is used to metabolize 5,000 calories (5 kcal) of energy. In this sense, volume of oxygen consumed can be converted to calories burned: 1mL O 2 = 5 cal. Absolute Oxygen consumption V O 2 : the rate of oxygen consumption, or the volume of oxygen consumed by the person per minute. The dot over the V to indicates a time derivative, so V O 2 is not the volume of oxygen consumed, but the rate at which oxygen is consumed [1]. The units of absolute V O 2 are ml min. Relative Oxygen consumption V O 2 : The rate of oxygen consumption per kilogram of body weight[1]. The units of relative V O 2 are ml kg/min. Metabolic Equivalents (METs): A simpler unit of power for clinical professionals. A MET is an abbreviation for a metabolic equivalent [of task]. One MET is equivalent to the relative Oxygen consumption rate at rest. [1] 1 MET = 3.5 ml kg min. Summary of useful units and conversions Profession Units of energy/work Units of power Conversion Physical Science Joule (J), calorie (cal) Watt (W) 1 W = 1 J/s Nutrition 1 Cal = 1 kcal kcal/time 1 cal = 4.19 J Physiology ATP or VO 2 (no dot) V O 2 1mL O 2 = 5 cal Medicine, Therapy Metabolic Equivalent (MET) 1 MET = 3.5 ml O 2 kg min Also, 1 mile = 1.609 km. 5. THE ACSM WALKING EQUATION Your V O 2 (and therefore power performance) can be directly measured in a laboratory with a respirator which measures how much oxygen is missing from the air you breathe out. Measurements like this have been analyzed using linear regression techniques such as we use in lab to create the American College of Sports Medicine (ACSM) walking equation. The walking equation is a model which can be used to estimate your relative V O 2 when walking at a certain speed on a certain grade. It is simply the sum of the power costs of moving horizontally, vertically, and just staying alive. V O 2 = (horizontal) + (vertical) + (resting) Adding these two components to the resting component (3.5 ml kg min) results in the ACSM walking equation. (Eq. 5) ml V O 2 ( kg min ) = v c h + v c v grade + c r
Where v is the walking speed in m/min and the coefficients c h, c h, and c h are given in the table below 1. c h = 0.1 ml m kg c v = 1.8 ml m kg ml c r = 3.5 min kg Work cost per horizontal meter Work cost per vertical meter Power cost of resting (1 MET) Note that grade is in decimal form: grade = rise/run. For example, a 15% grade in decimal form is 0.15. 6. DERIVING THE P-LAB WALKING EQUATION Physics students can recognize that the vertical part of the walking equation should depend on the vertical component of velocity. Consider the velocity vector of a person moving up an inclined plane of angle θ at a velocity v. Then the power cost due to the horizontal and vertical components of that velocity are horizontal = c h v cos θ vertical = c h v sin θ. Why does the ACSM walking equation not have cos θ? Why does it have grade instead of sin θ? Putting together the above pieces, we could write the P-Lab walking equation as Eq. 4. V O 2 (ml kg min) = c h v cos θ + c v v sin θ + c r (Eq. 6) V O 2 (ml kg min) = cos θ (v c h + v c v tan θ) + c r Notice that tan θ is equal to the grade of the slope ( rise over run, right?). Apparently, the ACSM walking equation assumes that the grades that people typically walk on is small enough that cos θ 1. On a flat grade then we obtain the same result as the ACSM walking equation. Suppose that we went for a walk on a 20% grade. Then cos θ = 0.981 1, so the ACSM walking equation is a decent way to estimate V O 2 (ml kg min) without the need for trigonometry. 7. BASIC EXERCISE PHYSIOLOGY The maximum heart rate (HR) max can be directly measured in a maximal test, calculated from a submaximal test, or simply estimated using a regression model. A simple and commonly-used model is (Eq. 7) (HR) max = 220 age. Heart rate reserve HRR is the difference between your maximum and resting heart rate Similarly, the maximal oxygen uptake reserve VO2R HRR = (HR) max (HR) rest. 1 The 2007 ACSM has 1.8 ml/min/m but these units are clearly incorrect.
VO2R = (V O 2 ) max (V O 2 ) rest, where (V O 2 ) max can be measured directly or calculated using a submaximal test such as the Rockport-1 mile test. The resting rate (V O 2 ) rest can be directly measured or estimated to be 3.5 ml/kg/s. While some fitness literature and fitness apps display HR as a percentage of (HR) max, a more accurate metric is %HRR as well as %VO2R, where %HRR = HR (HR) rest V O 2 (V O 2 ) HRR %VO2R = rest VO2R It turns out that HRR is linearly related to VO2R. Any person at rest is at 0% of HRR and 0% of VO2R. Any person at the anerobic threshold is at 100% of HRR and 100% of VO2R. For many people the relationship %HRR = %VO2R holds in between these extremes as well: (Eq. 9) %HRR = HR (HR) rest HRR = V O 2 (V O 2 ) rest VO2R = %VO2R Thus V O 2 (which requires expensive equipment) can be estimated by measuring HR (which only requires counting heart beats). PROCEDURE 1. YOUR ROCKPORT (V O 2 ) max AND KARVONEN THR CALCULATIONS 1. Follow the instructions at the end of this document for the Rockport 1-mile test. Record your data in Data Table 1. 2. How hard were you working 2?) Rate your relative perceived exertion (RPE) on a scale of 1-10 on the first row of the table in Data Table 3 ( Relate your HR to Power Expenditure ) 3. Estimate (V O 2 ) max using the Rockport 1-mile regression formula (Eq. 10) (V O 2 ) max = 132.853 0.0769 [weight(lbs)] 0.3877 [age(years)] + 6.315 [for males only] 3.2649 [time(minutes)] 0.1565 [heart rate (bpm)] 4. Determine your fitness level and the appropriate training intensity decimal for you using the table on the 5-5-5 fitness plan. 5. Calculate your target heart rate (THR) for optimal cardiovascular benefit using the Karvonen formula THR = HRR Intensity + (HR) rest 2 To say I was working hard is a statement about power (work per unit time), not work. To say I m working hard when climbing a hill really means that I m performing the work required to climb that hill relatively quickly, exerting more power but not performing more work.
2. RELATE YOUR HR TO POWER EXPENDITURE 1. Using a treadmill, adjust the incline to the maximum incline position. Many treadmills display a number indicating the incline, but you need to accurately measure the grade using a meter stick or tape measure. Then you ll know what that number means! Record in Data Table 2. 2. Choose a speed that would allow for you to comfortably carry on a conversation. Maintain that speed for 10 minutes, or until your HR reaches a stable warmed-up value. Record your speed, HR, and RPE in Data Table 3. 3. You are exercising at your own risk. Talk to the instructor if you don t believe that it would be a good idea to exercise at your THR, or exercise at a HR you are comfortable with. Find the speed that causes your HR to stay close to your THR for about 5 minutes. Record your speed, HR, and RPE. 4. Now make the treadmill flat and maintain your HR at your THR. Record your speed, HR, and RPE. You may find a spreadsheet helpful for the remaining calculations. Still, fill in the calculation sheet provided. 5. For each row in Data Table 3, calculate %HRR. Do you notice a relationship between %HRR and RPE? It turns out that many people often subjectively rate their power expenditure rather close to their actual %HRR. 6. Calculate your V O 2 for each row using the assumption that %HRR = %VO2R (Eq. 9). Were you exercising at your maximum HR? No? Then you are not calculating (V O 2 ) max here. 7. Calculate your power expenditure in METs. 8. Calculate the mechanical power required for each row and calculate your efficiency. Don t expect it to be high: your body was doing a lot of other things besides walking! TAKE IT TO THE NEXT LEVEL (FOR YOUR BENEFIT ONLY) 1. You can have your (V O 2 ) max measured directly. For instance, students at Southern Adventist University can ask to take this measurement in the human performance lab for free. 2. Would you like to learn more about using empirical models in sports medicine? You might start with Reference 1 (available in the campus library) or by perusing the freely-available ACSM 2011 standards [2]. 3. Would you like to read more about prescribing exercise to cardiac patients? You might start with Reference 3. REFERENCES [1] S. Glass, G.B. Dwyer, ed., ACSM's Metabolic Calculations Handbook, Lippincott Williams & Wilkins, Philadelphia, PA (2007). [2] C.E. Garber, et al., Quantity and Quality of Exercise for Developing and Maintaining Cardiorespiratory, Musculoskeletal, and Neuromotor Fitness in Apparently Healthy Adults: Guidance for Prescribing Exercise Medicine & Science in Sports & Exercise 43,1334 (2011). [3] M. Jetté, K. Sidney, G. Blümchen, Metabolic equivalents (METS) in exercise testing, exercise prescription, and evaluation of functional capacity. Clinical Cardiology, 8 (1990).
Data and Calculations 3. ROCKPORT (V O 2 ) max AND KARVONEN THR CALCULATIONS Data Table 1: (V O 2 ) max and THR calculations Time (min:s) HR max and HR rest Time (decimal min) Heart rate (bpm) (V O 2 ) max ( ml min /kg) HRR Training intensity THR 4. RELATE YOUR HR TO POWER EXPENDITURE What does the Incline number on the treadmill mean? Indicate on the drawing precisely where you measured rise and run (from where to where?) Data Table 2 Max Incline displayed Measured Rise Rat Race Treadmill Co. Run grade Treadmill walk *Use your judgement to exercise at your own risk. If it is not possible or advisable for you to reach your calculated target HR (THR), either use a lower HR within your comfort zone or talk to the Instructor. Data Table 3 Speed (mph) Rockport 0 Grade RPE HR (bpm) %HRR V O 2 ( ml min /kg) METs Effort (W/kg) Mech. Power (W/kg) Efficiency Incline Incline THR* Flat THR* 0
DISCUSSION 1. Cooter fell out of his tree-stand while hunting and broke both arms. Cooter has been training for the Mississippi Coast Half-marathon on November 24, but he can't run right now with two casts. Prescribe a reasonable treadmill walking exercise that will help him train at his goal of V O 2 = 38 ml min kg. You ll have to make reasonable choices, as in real life. 2. Prescribe a walking exercise program for Su-Yun, a 55-year old cardiac patient. Su-Yun prefers to walk outside on a level sidewalk downtown, where there are 20 city blocks per mile. She should exercise at 2.5 METs and walk for 30 minutes. Calculate her ideal walking speed and convert to the most convenient units for Su-Yun.
Name: PRELABORATORY INVESTIGATIONS Show your work. 1. What is the mechanical work required to move a 1.0-kg mass up a 10 degree inclined plane (such as a treadmill) which is 1.4 meters long? 2. A comfortable walking speed for many people is 1.4 m/s. What is the mechanical power required to move a 1.0-kg mass up a 10 degree inclined plane at this speed? Draw a vector diagram of the velocity, showing the components in the vertical and horizontal directions. Use SI units. 3. Before you got to sleep, put some kind of timer next to your bed. Measure your resting heart rate before rolling out of bed and record it here. If you forget, guess 60 bpm. 4. If you perform the Rockport 1-mile test (called the Duckwalk at SAU) on your own, you can skip that in lab. Record your time and heart rate here. 5. Come to lab dressed appropriately. The school gymnasium requires that you must wear closed-toed shoes. Jeans or riveted pockets are not allowed in the gym.
ROCKPORT ONE-MILE WALKING TEST INSTRUCTIONS Preparation: 1. Use your heart rate monitor if you have one. Otherwise, you can check out a heart rate monitor from the desk at the gym at SAU, or simply use a stopwatch. 2. Warm up thoroughly. Procedure: The goal is to get your heartrate up above 120 bpm, not to race. 1. Walk for one mile (4 laps on a quarter mile track, such as the one at SAU). Use the inside track for the most accurate results. 2. Stop walking after one mile (4 laps). 3. Record your time: : (minutes:seconds) 4. Record your heart rate: (bpm).