How long would it take a 1.5 x 10^5 kilogram airplane with engines that produce 100 megawatts of power to reach a speed of 250 m/s at an altitude of 12 kilometers is air resistance were negligible? If it actually takes 900 seconds, what is the power? Given this power what is the average force of air resistance if the airplane takes 1200 seconds? (Hint: you must find the distance the plane travels in 1200 seconds assuming constant acceleration.)
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To find the time it takes for the airplane to reach a certain speed and altitude, we can use the equation of motion:
v = u + at
where:
v = final velocity (250 m/s)
u = initial velocity (0 m/s, since the airplane starts from rest)
a = acceleration
t = time taken
Since the question mentions that air resistance is negligible, the only force acting on the airplane is the force produced by the engines. We can use Newton's second law of motion to relate the force, mass, and acceleration:
F = ma
Since power is given in the question, we can relate it to force and velocity:
Power (P) = Force (F) × velocity (v)
Now, we can find the acceleration using the equation:
v = u + at
Rearranging the equation, we get:
a = (v - u) / t
Substituting the given values:
a = (250 m/s - 0 m/s) / 900 s
a ≈ 0.278 m/s²
Now, we can calculate the force using Newton's second law:
F = ma
F = (1.5 × 10^5 kg) × 0.278 m/s²
F ≈ 41,700 N
To find the power, we can use the equation:
Power = Force × velocity
Plugging in the values:
Power = (41,700 N) × (250 m/s)
Power = 10,425,000 W = 10.425 MW
Therefore, the power of the engines is approximately 10.425 megawatts.
Next, we need to find the distance traveled by the airplane in 1200 seconds. Assuming constant acceleration, we can use the equation:
s = ut + (1/2)at²
Given:
u = 0 m/s (initial velocity)
t = 1200 s (time taken)
a = 0.278 m/s² (acceleration)
Plugging in the values:
s = (0 m/s) × (1200 s) + (1/2)(0.278 m/s²)(1200 s)²
s = 199,200 meters
Now, we can find the average force of air resistance. Since the given time is different from the previous case, we can't directly use the previously calculated acceleration. Instead, we can use the formula:
s = ut + (1/2)at²
Rearranging the equation, we get:
a = (2s - 2ut) / t²
Substituting the known values:
a = (2 × 199,200 m - 0 m) / (1200 s)²
a ≈ 2.7733 m/s²
Now, using Newton's second law, we can find the force:
F = ma
F = (1.5 × 10^5 kg) × 2.7733 m/s²
F ≈ 416,995 N
Therefore, the average force of air resistance is approximately 416,995 Newtons.