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# 1st year mechanics / physics question watch

1. Any help would be appreciated

An airplane in flight is subject to an air resistance force proportional to the square of its speed . 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 . 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 , so that the total air resistance force can be expressed by F=av^2 + b/v^2 , where a and b 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 a= 0.270 and b= 3.58×105 . In steady flight, the engine must provide a forward force that exactly balances the air resistance force.

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

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

I think i've figured out how to do the first part but cant do the second

the answers are 33.9ms^-1 and 25.8ms^-1 respectively

-taken from mastering physics
2. The maximum time will be the case where the power developed is a minimum. This uses the least amount of fuel per second and gives the longest time of flight.
Power is F x v, so multiply both sides by v and differentiate the RHS wrt v and equate to zero to find the minimum v.
I get the answer given using that method.
3. thanks for your help, I tried that but just realised I multiplied by v wrong

really appreciate the help

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