Question

An American press release describing the 1986 nonstop, round-the-world trip by the ultra-lightweight aircraft Voyager included the following data: flight distance: \(25,012 \mathrm{mi}\) flight time: 9 days, 3 minutes, 44 seconds fuel capacity: nearly 9000 lb fuel remaining at end of flight: \(14 \mathrm{gal}\) To the maximum number of significant figures permitted, calculate (a) the average speed of the aircraft in kilometers per hour (b) the fuel consumption in kilometers per kilogram of fuel (assume a density of \(0.70 \mathrm{g} / \mathrm{mL}\) for the fuel)

Short answer

a) The average speed of the aircraft is 186.21 kilometers per hour. b) The fuel consumption is 9.945 kg/km.

Step-by-step solution

Calculation For Average Speed

To find the average speed of the aircraft, we first need to convert the flight distance from mile to kilometer, and flight time from days to hours. 1 mile is equal to 1.60934 kilometers and 1 day is equal to 24 hours.The formula to calculate speed is given by:Speed = Distance / TimeConverted flight distance = \(25,012 \times 1.60934 = 40,230.77688 \, kilometers\)Converted flight time = \(9 \times 24 + \frac{3}{60} + \frac{44}{3600} = 216.0511 \, hours\)Now, we plug in these values into the formula to get the speed:Speed = \( \frac{40,230.77688}{216.0511} = 186.21 \, km/hr\) (to five significant digits)

Calculation For Fuel Consumption

To calculate fuel consumption, we need to know the weight of the consumed fuel in kilograms and then divide the total distance traveled by this weight. Before that, we have to convert the fuel weight from pounds to kilograms and volume from gallons to liters. 1 pound is equal to 0.453592 kilograms and 1 gallon is equal to 3.78541 liters.We are given the fuel capacity of 9,000 pounds, and we know that at the end of the flight, 14 gallons of fuel were remaining. We are also given that the fuel density is 0.70 g/mL.First, convert the initial fuel weight from pounds to kilograms: Initial fuel weight = \(9,000 \times 0.453592 = 4,082.328 \, kilograms\)Then, convert the remaining fuel volume from gallons to liters, and then to kilograms:Remaining fuel weight = \(14 \times 3.78541 \times 0.70 = 37.059758 \, kilograms\)Now, find the consumed fuel weight by subtracting remaining fuel weight from the initial fuel weight:Consumed fuel weight = \(4,082.328 - 37.059758 = 4,045.268242 \, kilograms\)Finally, divide the distance traveled by the consumed fuel weight to get the fuel consumption:Fuel consumption = \( \frac{40,230.77688}{4,045.268242} = 9.945 \, kg/km\) (to five significant digits)

Conclusion

In the calculation for average speed and fuel consumption, significant figures were maintained throughout all steps. After computing all values with appropriate conversions of units, final results were rounded off to 5 significant figures. Always ensure conversion of units to international standard units or as required by the problem before doing the calculation.

Key concepts

Average Speed Calculation

Calculating average speed is a fundamental aspect of many physics and chemistry problems. It helps us understand how fast an object is moving overall without considering specific points of the journey. To find average speed, you need two key pieces of information: distance traveled and the time it took to travel that distance. The formula to calculate average speed is:\[\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\]In this problem, the flight distance is given in miles and must be converted to kilometers. A handy conversion factor is that 1 mile is equivalent to 1.60934 kilometers. By multiplying the distance in miles by this factor, you convert the distance into kilometers, more suitable for standard scientific calculations.

Time conversion is equally important. Here, the flight time is given in days plus additional minutes and seconds. Since speed is typically calculated in hours, we must convert the entire duration into hours. Knowing that 1 day equals 24 hours, 1 minute is 1/60 of an hour, and 1 second is 1/3600 of an hour, allows us to perform this conversion accurately.

• Convert flight distance from miles to kilometers.
• Convert flight time from days, minutes, and seconds to hours.
• Use the formula to calculate the average speed.

By ensuring your units are consistent and by handling conversions carefully, you can calculate the average speed accurately.

Fuel Consumption Calculation

Understanding fuel consumption involves calculating how efficiently a vehicle uses fuel over a certain distance. For this, you need to know the total distance traveled and how much fuel was consumed during the trip. In the context of aviation, as in this exercise, calculations need to be accurately performed due to the considerable distances and fuel weights. You begin by calculating the weight of fuel consumed. Initially, convert the fuel’s total weight in pounds to kilograms, which is the standard metric unit. The conversion factor is essential here, with 1 pound being equal to 0.453592 kilograms. At the end, the remaining fuel needs to be subtracted from the initial fuel amount to find the actual fuel consumed. The remaining fuel is converted from gallons to liters, since 1 gallon equals 3.78541 liters, and then into kilograms using the given density of the fuel (0.70 g/mL).

• Convert the initial fuel weight from pounds to kilograms.
• Convert remaining fuel volume to liters, then to kilograms.
• Calculate consumed fuel by subtracting remaining from initial.
• Calculate fuel consumption using distance traveled and consumed fuel weight.

Through these steps, you can derive the fuel consumption in terms of weight over distance, providing insight into the fuel efficiency of the aircraft.

Unit Conversion in Chemistry

Unit conversion is a vital skill in chemistry and physics, ensuring various quantities are accurately translated into useful metrics. Converting units helps streamline calculations and avoids errors when dealing with international data or scientific standards. In our context, we see multiple types of conversions: 1. **Distance Conversion:** Distances are often given in miles, particularly in regions like the United States. However, many scientific formulas use kilometers. Thus, miles must be converted using the conversion factor where 1 mile equals 1.60934 kilometers. 2. **Time Conversion:** Time given in days, minutes, and seconds needs to be consistently formatted, especially for speed calculations where hours are ideal. The conversion is straightforward, using basic arithmetic: - 1 day = 24 hours - 1 minute = 1/60 hour - 1 second = 1/3600 hour 3. **Weight and Volume Conversion:** Chemistry often employs metric units. For weight, pounds are converted to kilograms using 1 pound = 0.453592 kilograms. For volume, gallons are shifted to liters with 1 gallon = 3.78541 liters. When density is involved, like in fuel calculations, it’s necessary to convert to mass (kilograms).

• Recognize the need for conversion based on given units.
• Use appropriate conversion factors consistently.
• Maintain accuracy in calculations to preserve significant figures.

Accurate unit conversion supports precise calculations, ensuring results are scientifically reliable and easily interpretable.