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 can use the formula for work done. The work done in this case is equal to the change in kinetic energy, which is given by (1/2)mv^2. We can calculate the work done by using the following steps:

1. Calculate the change in kinetic energy:
ΔKE = (1/2)mv^2 - (1/2)mv₀^2
Here, m is the mass of the plane (200,000 kg), and v and v₀ are the final and initial velocities, respectively.
We know the initial velocity is zero, v₀ = 0.

2. Find the total energy required to get the plane to the takeoff speed:
E = ΔKE / Efficiency of engines
The efficiency of the engines is given as 22.0% or 0.22.

3. Determine the amount of heat produced by the fuel:
Heat produced = E / Energy produced per gallon of jet fuel

Now, let's calculate part (a) using the given values:
Mass of the plane (m) = 200,000 kg
Takeoff speed (v) = 55.0 m/s
Efficiency of the engines = 22.0% or 0.22
Energy produced per gallon of jet fuel = 1.30 × 10^8 J

(a) How many gallons of fuel are used to get the plane from rest to the takeoff speed of 55.0 m/s?

1. Calculate the change in kinetic energy:
ΔKE = (1/2)(200,000 kg)(55.0 m/s)^2 - (1/2)(200,000 kg)(0 m/s)^2
ΔKE = (1/2)(200,000 kg)(55.0 m/s)^2

2. Find the total energy required to get the plane to the takeoff speed:
E = ΔKE / Efficiency of engines
E = [(1/2)(200,000 kg)(55.0 m/s)^2] / (0.22)

3. Determine the amount of heat produced by the fuel:
Heat produced = E / Energy produced per gallon of jet fuel
Heat produced = [E] / (1.30 × 10^8 J)

So, by substituting the values into the respective formulas, we can find the number of gallons of fuel used.

(a)

mv₁²/2=0.22•N₁•Q
N₁=mv₁²/2•0.22•Q =
=2•10⁵•55²/2•0.22•1.3•10⁸ =10.6 gal
(b)
0.22•N₂•Q= mv₂²/2 -mv₁²/2 +mgh,
N₂ = m(v₂² -v₁² +2gh)/2•0.22•Q=
=2•10⁵(250²- 55² +2•9.8•10⁴)/2•0.22•1.3•10⁸=
=900.3 gal
(c) Q₁ = N₁•Q=10.6•1.3•10⁸=1.37•10⁹ J
Waste Q₁ =0.78•1.37•10⁹=1.07•10⁹ J
Q₂=N₂•Q= 900.3•1.3•10⁸ =1.17•10¹¹ J
Waste Q₂= 0.78•1.17•10¹¹ = 9.13•10¹⁰ J