Question

Human-Powered Flight Human-powered aircraft require a pilot to pedal, as on a bicycle, and to produce a sustained power output of about \(0.30 \mathrm{hp}(1 \mathrm{hp}=746 \mathrm{~W})\). The Gossamer Albatross flew across the English Channel on June 12,1979 , in \(2 \mathrm{~h} 49 \mathrm{~min}\). (a) How much energy did the pilot expend during the flight? (b) How many candy bars (280 Cal per bar) would the pilot have to consume to be "fueled up" for the flight? Note that a nutritional calorie (1 Cal) is equivalent to 1000 calories (1000 cal) as defined in physics. In addition, the conversion factor between calories and joules is as follows: \(1 \mathrm{Cal}=1000 \mathrm{cal}=1 \mathrm{kcal}=4186 \mathrm{~J} .\)

Short answer

The pilot expended 2268432 J of energy and would need 2 candy bars.

Step-by-step solution

Convert Power to Watts

First, convert the human power from horsepower (hp) to watts. Given that 1 hp = 746 W, we have:\[ 0.30 \text{ hp} = 0.30 \times 746 \text{ W} = 223.8 \text{ W} \]

Convert Time to Seconds

The flight time is given as 2 hours and 49 minutes. First, convert 2 hours to minutes:\[ 2 \text{ hours} = 2 \times 60 = 120 \text{ minutes} \]Then add the additional 49 minutes:\[ 120 \text{ minutes} + 49 \text{ minutes} = 169 \text{ minutes} \] Convert minutes to seconds:\[ 169 \text{ minutes} \times 60 = 10140 \text{ seconds} \]

Calculate Energy in Joules

Energy is the product of power and time. The energy expended by the pilot is:\[ \text{Energy} = \text{Power} \times \text{Time} = 223.8 \text{ W} \times 10140 \text{ s} \]Calculating the above:\[ \text{Energy} = 2268432 \text{ J} \]

Convert Joules to Calories

Using the conversion factor \( 1 \text{ Cal} = 4186 \text{ J} \), convert joules to calories:\[ \text{Calories} = \frac{2268432 \text{ J}}{4186 \text{ J/Cal}} \approx 541.8 \text{ Cal} \]

Calculate Number of Candy Bars

Each candy bar provides 280 Cal. Divide the total calories by the calories per candy bar:\[ \text{Number of candy bars} = \frac{541.8 \text{ Cal}}{280 \text{ Cal/bar}} \approx 1.936 \text{ bars} \] Since partial candy bars are not possible, the pilot would require at least 2 candy bars.

Key concepts

Energy Conversion

In human-powered flight, energy conversion is crucial. The pilot converts chemical energy from food into mechanical energy through pedaling. This energy is what allows the aircraft to stay in motion. The power generated by the pilot is initially measured in horsepower, a common unit for mechanical power. To better understand and calculate energy use, we convert this power into watts, which is the standard international unit of power. To convert from horsepower to watts, we use the conversion factor: 1 horsepower = 746 watts. Therefore, for our exercise, a power output of 0.30 hp is equivalent to 223.8 watts. This conversion allows us to use a consistent metric (watts), making power and energy calculations more straightforward. By understanding these energy conversion processes, we can more efficiently quantify and optimize the energy use in human-powered vehicles.

Power and Energy Calculations

Power and energy calculations are essential to assessing energy expenditure during human-powered flights. Power, measured in watts, is a rate of doing work or energy per unit time. Therefore, to find the total energy expended, we multiply power by time. In our scenario, the flight lasted 2 hours and 49 minutes, which we convert to 10140 seconds for easy calculations. With the power output of 223.8 watts, the energy expended is 2268432 joules (J), calculated by multiplying the power in watts by the time in seconds. After determining the energy in joules, we often convert it to nutritional units (calories) to relate energy expenditure to food consumption. Using the conversion 1 Calorie (Cal) = 4186 J, the 2268432 J equates to approximately 541.8 Cal.

Nutritional Energy Requirements

Understanding nutritional energy requirements helps relate physical activities to dietary intake. In this exercise, after converting the energy expenditure from joules to calories, we calculate how much food the pilot needs before the flight. Every activity requires a specific amount of nutritional energy. For our pilot, the required energy to maintain flight for 2 hours and 49 minutes is approximately 541.8 Cal. Given that a typical candy bar contains about 280 Cal, the pilot would need energy from food equal to roughly two candy bars to meet these nutritional demands. This means the energy ingested through food should match the energy expended during the activity to ensure the pilot has the endurance needed for the flight. Through understanding and applying these calculations, one can effectively balance nutritional intake with energy requirements for various physical endeavors.