An airplane in flight is subject to an air resistance force proportional to the

Question

An airplane in flight is subject to an air resistance force proportional to the square of its speed v. But there is an additional resistive force because the airplane has wings. Air flowing over the wings is pushed down and slightly forward, so from Newton's third law the air exerts a force on the wings and airplane that is up and slightly backward (Figure 1) . The upward force is the lift force that keeps the airplane aloft, and the backward force is called induced drag. At flying speeds, induced drag is inversely proportional to v2, so that the total air resistance force can be expressed by Fair=v2+/v2, where and are positive constants that depend on the shape and size of the airplane and the density of the air. To simulate a Cessna 150, a small single-engine airplane, use = 0.330Ns2/m2 and = 3.46Ý105Nm2/s2 . In steady flight, the engine must provide a forward force that exactly balances the air resistance force.

a.Calculate the speed at which this airplane will have the maximum range (that is, travel the greatest distance) for a given quantity of fuel.

b.Calculate the speed for which the airplane will have the maximum endurance (that is, will remain in the air the longest time).

An airplane in flight is subject to an air resistance force proportional to the square of its speed v. But there is an additional resistive force because the airplane has wings. Air flowing over the wings is pushed down and slightly forward, so from Newton's third law the air exerts a force on the wings and airplane that is up and slightly backward (Figure 1) . The upward force is the lift force that keeps the airplane aloft, and the backward force is called induced drag. At flying speeds, induced drag is inversely proportional to v2, so that the total air resistance force can be expressed by Fair=??±v2+???²/v2, where ??± and ???² are positive constants that depend on the shape and size of the airplane and the density of the air. To simulate a Cessna 150, a small single-engine airplane, use ??± = 0.330Nâ?½?½s2/m2 and ???² = 3.46???½105Nâ?½?½m2/s2 . In steady flight, the engine must provide a forward force that exactly balances the air resistance force. a.Calculate the speed at which this airplane will have the maximum range (that is, travel the greatest distance) for a given quantity of fuel. b.Calculate the speed for which the airplane will have the maximum endurance (that is, will remain in the air the longest time).

Explanation / Answer

to have maximum range for a certain fuel, you might convince yourself this happens when the resistance force is at its minumum

when it is minimum ? using calculus this will be true when df/dv = 0 and d/dv df/dv = positive number

so differneitate the force and if you get more solutions for velocity chose the one where the double derivative is positive

to remain in the air for the longest time , you are basically saying maximimise time, so it will take much longer to pass through, which means i think the resistance force should be maximum - i may be wrong here

so do df/dv = 0 and d/dv df/dv = negative number this will give you the velocity

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