The aeroplane of mass M requires a speed v for take-off. The length of the run-way is s and the coefficient of friction between the tyres and ground is µ. Assuming that the plane accelerates 

The acceleration a required to impart a speed v in a distance s is given by

${\mathrm{v}}^2-{\mathrm{u}}^2=2\mathrm{as}\text{ or }{\mathrm{v}}^2=2\mathrm{as}(\because \mathrm{u}=0)$

$\text{ or }\mathrm{a}={\mathrm{v}}^2/2\mathrm{s}$

The force F₁ needed to produce this acceleration is

${\mathrm{F}}_1=\mathrm{M}\times \mathrm{a}=\mathrm{M}\left({\mathrm{v}}^2/2\mathrm{s}\right)$

The force F₂ needed to overcome the friction is

${\mathrm{F}}_2=\mathrm{\mu Mg}$

$\begin{array}{rl}\therefore \text{Total force needed}& ={\mathrm{F}}_1+{\mathrm{F}}_2\\ & =\mathrm{M}(\frac{{\mathrm{v}}^2}{2\mathrm{s}}+\mathrm{\mu g})\end{array}$