Question

Flight Human-powered aircraft require a pilot to pedal, as on 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 \(2 \mathrm{~h} \mathrm{} 49 \mathrm{~min}\). (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 ( \(1 \mathrm{Cal}\) ) is equivalent to 1000 calories ( \(1000 \mathrm{cal}\) ) 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

(a) 2261532 J, (b) 2 candy bars are needed.

Step-by-step solution

Convert Horsepower to Watts

The power output in horsepower is given as 0.30 hp. Since 1 hp = 746 W, we can convert this to watts by multiplying: \[0.30 \, \text{hp} \times 746 \, \frac{\text{W}}{\text{hp}} = 223.8 \, \text{W}\]

Convert Flight Time to Seconds

The duration of the flight is 2 hours and 49 minutes. First, convert the time to minutes: \[2 \, \text{hours} = 2 \times 60 = 120 \, \text{minutes}\]Then, add the remaining minutes: \[120 \, \text{minutes} + 49 \, \text{minutes} = 169 \, \text{minutes}\]Now, convert minutes to seconds: \[169 \, \text{minutes} \times 60 \, \frac{\text{seconds}}{\text{minute}} = 10140 \, \text{seconds}\]

Calculate Total Energy Expended in Joules

The energy expended is calculated using the formula: \[\text{Energy} = \text{Power} \times \text{Time}\]Substitute the power in watts and the time in seconds: \[223.8 \, \text{W} \times 10140 \, \text{seconds} = 2261532 \, \text{Joules}\]

Convert Energy from Joules to Calories

Since 1 Cal (nutritional calorie) is equivalent to 4186 Joules, convert the energy from joules to Calories:\[2261532 \, \text{J} \div 4186 \, \frac{\text{J}}{\text{Cal}} = 540.31 \, \text{Cal}\]

Calculate Number of Candy Bars Needed

Each candy bar provides 280 Cal. The total energy in Calories is 540.31 Cal. Calculate the number of candy bars:\[\frac{540.31 \, \text{Cal}}{280 \, \frac{\text{Cal}}{\text{bar}}} = 1.93 \, \text{bars}\]Since the pilot cannot consume a fraction of a candy bar practically, we round up to the nearest whole number, which is 2 bars.

Key concepts

Power Output

Power output is crucial when discussing energy conversion. It is the rate at which work is done or energy is transferred. In the context of human-powered flight, it refers to the amount of energy that a pilot must continuously generate to keep an aircraft airborne. For the Gossamer Albatross, this output was 0.30 horsepower (hp). By converting horsepower to watts, which is an easier unit to handle in calculations, we find the pilot generates approximately 223.8 watts of power. This conversion is essential because it sets the stage for calculating the total energy expended over time.

Calories to Joules Conversion

Converting calories to joules is an important step in understanding energy expenditure in scientific terms. A nutritional calorie, often written as Cal, is equivalent to 1000 calories in the scientific context. This means 1 Cal is equal to 4186 joules. Since energy calculations often involve joules in physics, being able to switch between these units is vital for calculations like determining the energy expended during a flight.

Human-Powered Flight

Human-powered flight is a fascinating concept where an aircraft is propelled solely by the physical effort of a pilot. This kind of flight requires a continuous and sustainable power output, as was demonstrated by the Gossamer Albatross. For its successful flight across the English Channel, the pilot pedaled for almost three hours, demanding efficient energy usage and tremendous physical endurance. This remarkable achievement highlights the importance of energy management in such flights, using renewable, human-generated energy.

Energy Expenditure Calculation

Calculating energy expenditure involves translating the power output over a given period into an amount of energy used. For the pilot of the Gossamer Albatross, this means considering how the continuous pedaling over 10140 seconds (the duration of the flight) results in a total energy usage of 2261532 joules. The next step was to convert this into Calories (noting 1 Cal = 4186 J), resulting in approximately 540.31 Cal. This calculation helps determine how much physical energy, equivalent to eating certain foods like candy bars, the pilot needed to complete the journey, highlighting the connection between energy consumption and physical performance.