When its engine of power 75.0 kW is generating full power, a small single-engine airplane with mass 700 kg gains altitude at a rate of 2.60 m/s .
A)What fraction of the engine power is being used to make the airplane climb? (The remainder is used to overcome the effects of air resistance and of inefficiencies in the propeller and engine.)
Compare the rate of increase in potential energy with the engine power. The ratio will be the fraction that you are being asked to determine.
The rate of increase in potential energy is M g dh/dt.
dh/dt is the rate of increase in altitude, 2.60 m/s. m is the mass and g is the acceleration of gravioty.
To determine the fraction of the engine power being used to make the airplane climb, we need to compare the rate of increase in potential energy with the engine power.
The rate of increase in potential energy is given by the equation:
Rate of increase in potential energy = mass (m) * acceleration due to gravity (g) * rate of increase in altitude (dh/dt)
We are given that the rate of increase in altitude (dh/dt) is 2.60 m/s, the mass (m) of the airplane is 700 kg, and the acceleration due to gravity (g) is approximately 9.8 m/s^2.
Substituting these values into the equation, we have:
Rate of increase in potential energy = (700 kg) * (9.8 m/s^2) * (2.60 m/s)
Now, let's calculate the rate of increase in potential energy:
Rate of increase in potential energy = (700 kg) * (9.8 m/s^2) * (2.60 m/s) = 17996 J/s
The engine power is given as 75.0 kW, which can be converted to joules per second (J/s) by multiplying by 1000:
Engine power = 75.0 kW * 1000 = 75000 J/s
Finally, we can find the fraction of the engine power being used to make the airplane climb by dividing the rate of increase in potential energy by the engine power:
Fraction of engine power used for climbing = Rate of increase in potential energy / Engine power
Fraction of engine power used for climbing = 17996 J/s / 75000 J/s
Fraction of engine power used for climbing ≈ 0.2399
Therefore, approximately 0.2399 or 23.99% of the engine power is being used to make the airplane climb. The remaining power is used to overcome air resistance and inefficiencies in the propeller and engine.