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 ll 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 m/s2-9.8~\mbox{m/s}^2.
  • The base mass m0m_0 of the helicopter without the battery is 50 g50~\mbox{g}.
  • The battery is a uniform cube with side length ll and density ρ=4000 kg/m3\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 l2.7l^{2.7}.
  • The emitted voltage from the battery is constant and does not change with ll. 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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