(a) Suppose 10 astronauts are sent to Mars and then back to Earth. The average power usage of
each astronaut is 10,000,000 joules per day. What is the average power usage of each astronaut
in watts and horsepower?
(b) It will take approximately 2 years for the astronauts to make the round trip. The average astronaut
has a mass of 70.0 kg. Assume the astronauts are on a fat-free diet. Carbohydrates and proteins
on average produce approximately 17,300,000 joules of energy per kilogram. How much mass of
food do the astronauts need to take with them? Note that we are assuming that 100% of the
food calories that are eaten are absorbed and used by the body. This is actually not true. A
person’s metabolic efficiency” is the percentage of calories eaten that are actually used; the rest
are eliminated by the body. Metabolic efficiency varies considerably from person to person. This
means that the mass of food calculated is an underestimate of what would be needed.
(c) 1 food calorie (Cal) = 1 kilocalorie (kcal) = 1,000 calories (cal) = 4,184.5 joules of food requires
approxiamtely 0.000248 kg of oxygen for the body to metabolize the food. How many kilograms
of oxygen are required for the trip?
(d) Assume it costs $10,000,000 to send 1 kg to Mars. How much will it cost to send the astronauts,
their food and oxygen to Mars?
(e) Assume 15% of the energy goes into mechanical energy, how much mechanical work will the 10
astronauts do over the 2 years?
I will be happy to check your work.
(a) To find the average power usage of each astronaut in watts, we need to convert the energy usage from joules per day to watts.
Since power is defined as energy divided by time, we divide the energy usage (10,000,000 joules per day) by the number of seconds in a day (24 hours * 60 minutes * 60 seconds). This gives us the power usage in watts.
Watts = Energy (joules) / Time (seconds)
Watts = 10,000,000 joules / (24 * 60 * 60 seconds)
= 115.74 watts
Therefore, the average power usage of each astronaut is approximately 115.74 watts.
To convert this value to horsepower, we can use the conversion factor:
1 horsepower (hp) = 745.7 watts
Horsepower = Watts / 745.7
Horsepower = 115.74 watts / 745.7
≈ 0.1554 horsepower
Therefore, the average power usage of each astronaut is approximately 0.1554 horsepower.
(b) To find the mass of food required for the astronauts, we need to calculate the energy required for the round trip and divide it by the energy produced per kilogram of carbohydrates and proteins.
Energy required for the round trip = Energy usage per day (10,000,000 joules) * Number of days (365 days/year * 2 years) * Number of astronauts (10)
Energy required for the round trip = 10,000,000 joules/day * 365 days/year * 2 years * 10 astronauts
= 7.3 x 10^12 joules
Energy produced per kilogram of carbohydrates and proteins = 17,300,000 joules/kg
Mass of food required = Energy required for the round trip / Energy produced per kilogram of carbohydrates and proteins
Mass of food required = 7.3 x 10^12 joules / 17,300,000 joules/kg
≈ 421,686 kg
Therefore, approximately 421,686 kg of food is needed for the astronauts.
(c) To find the kilograms of oxygen required for the trip, we need to calculate the amount of oxygen required to metabolize the food.
Mass of food in kilocalories = Energy required for the round trip / 4,184.5 joules/calorie
Mass of food in kilocalories = 7.3 x 10^12 joules / 4,184.5 joules/kcal
≈ 1.745 x 10^9 kcal
Mass of oxygen required = Mass of food in kilocalories * 0.000248 kg of oxygen/kcal
Mass of oxygen required = 1.745 x 10^9 kcal * 0.000248 kg of oxygen/kcal
≈ 432,760 kg of oxygen
Therefore, approximately 432,760 kg of oxygen is required for the trip.
(d) To calculate the cost of sending the astronauts, their food, and oxygen to Mars, we need to multiply the total mass by the cost per kilogram.
Cost to send 1 kg to Mars = $10,000,000
Total cost = (Mass of astronauts + Mass of food + Mass of oxygen) * Cost to send 1 kg to Mars
Total cost = (10 astronauts * 70.0 kg/astronaut + 421,686 kg + 432,760 kg) * $10,000,000
Total cost ≈ $88,247,460,000,000
Therefore, it will cost approximately $88,247,460,000,000 to send the astronauts, their food, and oxygen to Mars.
(e) To calculate the mechanical work done by the 10 astronauts over the 2-year period, we need to multiply the average power usage by the time.
Mechanical work = Power (watts) * Time (seconds)
Power = 115.74 watts (average power usage of each astronaut)
Time = 2 years * 365 days/year * 24 hours/day * 60 minutes/hour * 60 seconds/minute
Mechanical work = 115.74 watts * (2 * 365 * 24 * 60 * 60 seconds)
Mechanical work ≈ 7.30 x 10^12 joules
Therefore, the 10 astronauts will do approximately 7.30 x 10^12 joules of mechanical work over the 2-year period.