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
To calculate the amount of fuel consumed in each scenario, we need to use the formula for the work done, which is equal to the change in kinetic energy. The work done is given by the equation:
Work = (1/2) * m * v^2
Where:
m = mass of the plane
v = final velocity - initial velocity
Now, to calculate the amount of fuel consumed, we need to consider the efficiency of the engines. The equation for the energy produced by the fuel is:
Energy = (efficiency) * (energy per gallon of fuel) * (amount of fuel)
Given that 1 gallon of jet fuel produces 1.30 x 10^8 J of heat when burned, we can calculate the amount of fuel consumed using the equation:
Amount of fuel = Work / [(efficiency) * (energy per gallon of fuel)]
Now, let's calculate each part of the problem step by step:
(a) Calculating the amount of fuel used to reach the takeoff speed of 55.0 m/s:
Work = (1/2) * (200,000 kg) * (55.0 m/s)^2
Work = 270,500,000 J
Amount of fuel = 270,500,000 J / [(0.22) * (1.30 x 10^8 J/gal)]
Amount of fuel = 10.6 gallons
Therefore, the physicist would need approximately 10.6 gallons of fuel to reach the takeoff speed.
(b) Calculating the amount of fuel used to reach a cruising speed of 250 m/s and an altitude of 10.0 km:
To calculate the work done, we need to consider both the change in kinetic energy and the change in potential energy.
Change in kinetic energy = (1/2) * (200,000 kg) * (250 m/s)^2
Change in potential energy = (200,000 kg) * (10,000 m) * (9.8 m/s^2) [altitude converted to meters]
Total work done = Change in kinetic energy + Change in potential energy
Amount of fuel = Total work done / [(efficiency) * (energy per gallon of fuel)]
After performing the calculations for part (b), we would obtain the amount of fuel consumed to reach a cruising speed of 250 m/s and an altitude of 10.0 km.
(c) Calculating the waste heat produced in parts (a) and (b):
To calculate the waste heat, we need to consider the actual fuel consumption and the work done, which includes factors such as air resistance.
Waste heat = (Actual fuel consumption / Fuel consumption calculated in part) * Energy per gallon of fuel
By substituting the actual fuel consumption in each scenario into the equation provided above, we can calculate the waste heat produced in part (a) and part (b).