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 byv2−u2=2as or v2=2as (∵u=0) or a=v2/2sThe force F1 needed to produce this acceleration isF1=M×a=Mv2/2sThe force F2 needed to overcome the friction isF2=μMg∴ Total force needed=F1+F2 =M(v22s+μg)