An electric vehicle of mass ##m## moves along the ##0-x## axis according to the law:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]m\frac{\mathrm{d} v(t)}{\mathrm{d} t} = T(t)-\mu mg-kv(t)^{2}[/itex] (ma = Thrust - Friction - Drag)

It is known that:

[itex]P(t) = T(t)v(t)[/itex] (Power = Thrust * Speed)

Find the thrust ##T(t)## in such a way that the vehicle be able to reach a destination as far from origin as possible with a given amount of energy, ##E##, stored in its batteries, in a time [itex]\tau < \tau_{0}[/itex].

[itex]E = \int_{0}^{\tau}P(t)dt[/itex]

The quantity that has to be maximized is the distance, ##d##, covered:

[itex]d = \int_{0}^{\tau}v(t)dt[/itex]

##E, m, \tau_{0}, g = 9.81 m/s^2, \mu, k##, the last two being the friction and drag coefficients, are all parameters with given values.

I have no idea how to solve such a problem. Some suggestions about finding ##T(t)##, in Mathcad for instance, would be welcomed.

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# Find the best T(t) from a differential eq. and conditions

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