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

Excerpt
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).