Long lasting kid fun

Toy helicopters with rechargeable batteries fly for a few minutes on a single charge. Manufacturers want to choose the right size of battery to achieve the longest flight time between charges. A larger battery stores more energy, but also increases the mass of the helicopter so it takes more energy to keep it in the air. Our question is, for a helicopter and battery type that behave as specified in the assumptions below, what is the linear size \(l\) of the battery in mm that will maximize the time the helicopter can hover in place?

Details and assumptions

  • The acceleration of gravity is \(-9.8~\mbox{m/s}^2\).
  • The base mass \(m_0\) of the helicopter without the battery is \(50~\mbox{g}\).
  • The battery is a uniform cube with side length \(l\) and density \(\rho= 4000 ~\mbox{kg/m}^3\).
  • The total energy contained in the battery does not quite scale as the volume does, but instead scales as \(l^{2.7}\).
  • The emitted voltage from the battery is constant and does not change with \(l\). All that changes is how long the battery lasts. Therefore the rotors spin and push air in the same way no matter what the battery size is.
  • Assume the mass of the air pushed by the rotors is a constant value, independent of rotor speed. In an actual helicopter the mass increases with speed, effectively from Bernoulli's equation, but that's more complication than we want to go into.

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