While taking a plane flight to a meeting, a physicist decides to do a mental calculation of how much fuel the plane would consume in the absence of friction, assuming its engines have an efficiency of 22.0% and the mass of the plane is 200,000 kg. Each gallon of jet fuel produces 1.30 multiplied by 108 J of heat when burned.
(a) How many gallons of fuel are used to get the plane from rest to the takeoff speed of 55.0 m/s?
(b) How many gallons to get to a cruising speed of 250 m/s and an altitude of 10.0 km?
(c) How much waste heat is produced in each case(in part a and b)? (Both the actual fuel consumption and heat produced will be greater because of air resistance.)
I know how to do the first one.
(0.5*200000*55^2)/[(1.30*10^8)*0.22]
=10.6 gal
And part of (c)
Q waste=1.3*10^8*10.6-0.5*200000*55^2=1.08*10^9 J
To solve part (a) of the problem, you correctly used the equation:
\( \text{Fuel Used} = \frac{0.5 \times \text{mass} \times \text{velocity}^2}{\text{efficiency} \times \text{energy per gallon}} \)
Substituting the given values:
\( \text{Fuel Used} = \frac{0.5 \times 200,000 \times 55^2}{0.22 \times 1.3 \times 10^8} \)
Evaluating the expression will give you the answer: 10.6 gallons.
Now, let's move on to part (b) of the problem. We need to find the amount of fuel required to reach a cruising speed of 250 m/s and an altitude of 10.0 km. However, we don't have enough information to calculate this directly.
To solve this, we need to break down the problem into two parts:
1. The fuel required to reach the cruising speed from the initial takeoff speed.
2. The additional fuel required to gain altitude.
For the first part, let's calculate the fuel required to reach the cruising speed of 250 m/s:
\( \text{Fuel Used} = \frac{0.5 \times \text{mass} \times (\text{cruising speed}^2 - \text{takeoff speed}^2)}{\text{efficiency} \times \text{energy per gallon}} \)
Substituting the given values:
\( \text{Fuel Used} = \frac{0.5 \times 200,000 \times (250^2 - 55^2)}{0.22 \times 1.3 \times 10^8} \)
Evaluate this expression to find the fuel required for the first part.
For the second part, we need to calculate the fuel required to gain altitude. Unfortunately, we don't have enough information about the plane's climb rate or the time it takes to reach the desired altitude. Therefore, we cannot accurately calculate the amount of fuel required to gain altitude.
Moving on to part (c), to calculate the waste heat produced, we need to subtract the work done on the plane (fuel energy) from the total energy produced by burning the fuel. The equation is:
\( \text{Waste Heat} = \text{Energy Per Gallon} \times \text{Fuel Used} - \text{Work Done} \)
For part (a), we have the formula:
\( \text{Waste Heat} = \text{Energy Per Gallon} \times \text{Fuel Used} - 0.5 \times \text{mass} \times \text{takeoff speed}^2 \)
Substituting the given values:
\( \text{Waste Heat} = (1.3 \times 10^8 \times \text{Fuel Used}) - (0.5 \times 200,000 \times 55^2) \)
Evaluate this expression to find the waste heat produced in part (a).
Similarly, for part (b), you can use the same formula to calculate the waste heat:
\( \text{Waste Heat} = (1.3 \times 10^8 \times \text{Fuel Used}) - (0.5 \times \text{mass} \times \text{cruising speed}^2) \)
Substitute the fuel used for the first part of part (b) into the equation and evaluate it to find the waste heat produced.