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The maximum Force possible for a vehicle to obtain a maximum acceleration without slipping is Fsmax.

Proof:- As shown in figure, if f = fsmax then we know that F and f are couple forces. So, the tyre will roll instead of slipping and F = fsmax.

If f = fk then acceleration will produced because if the vehicle is moving the friction acting on the vehicle (fk) is always less than applied force (F) or maximum static friction (fsmax).

fsmax > fk

F > fk                           since F = fsmax

To find the acceleration we know that

F - fk = ma

fsmax - fk = ma

musmax *mg - muk *mg = ma

mg (musmax - muk ) = ma

Or

a = g(musmax - muk)