Flying high

Classical Mechanics Level 4

An airplane, when in flight, is subjected to drag that is proportional to its speed $v$ . But air flowing over the wings also pushes it downward. Given this consideration, we can model the total air resistance force on a plane as $F_f = av^2 + bv^{-2}$ for some constants $a$ and $b$ which depend on the airplane.

Consider a plane that is in a steady flight at which the engine must provide an equal and opposite force to the resistance force. The speed at which the airplane flies the longest distance given a specific amount of fuel can be expressed as $v = \left(K \times \frac{b}{a}\right)^\frac{1}{M}$ where $K$ and $M$ are positive integers. What is the value of $K + M$ ?

The answer is 5.

1 solution

Azher Ferrer
Nov 15, 2014

For the airplane to travel the longest distance given a specific amount of fuel to be consumed, the total air resistance must be minimized .

To minimize the total air resistance acting on the airplane, take the first derivative of the total air resistance with respect to the speed and set it to zero :

$\frac{dF_f}{dv}\ =\ 0 \ =\ 2av-2bv^{-3}$

Solving for v, we get $v=(\frac{b}{a})^{\frac{1}{4}}$ .

Equating the expression for v in $v=(K\times\frac{b}{a})^{\frac{1}{M}}$ , we have $K=1$ and $M=4$ . Therefore, the final answer is $K+M=\boxed 5$ .

exactly the same concept

Anirban Ghosh - 6 years, 7 months ago

nice solution fully explained sir @azher Ferrer

Mardokay Mosazghi - 6 years, 5 months ago

Flying high

The answer is 5.

1 solution

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