# 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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