# An airplane in flight is subject to an air resistance force proportional to the square of its speed v. But there

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at wch ts airplane will have the maximum range (that is, travel the greatest distance) for a given quantity of fuel. 2. Calculate the speed (in km/h) for wch the airplane will have the maximum endurance (that is, 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 trd 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 v

2

, so that the total air

resistance force can be expressed by

Fair = av2 + b/v2

where a and b are positive constants that depend on the shape and size of

the airplane and the density of the air.

For a Cessna 150, a small single-engine airplane, a = 0.30 N.s2

/m2 and

b = 3.5×105 N.m2

/s

2

. In steady flight, the engine must provide a forward force

that exactly balances the air resistance force.

1. Calculate the speed (in km/h) at wch ts airplane will have the maximum range (that is, travel the greatest distance) for a given quantity of

fuel.

2. Calculate the speed (in km/h) for wch the airplane will have the maximum endurance (that is, remain in the air the longest time).

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