Suppose we consider a particular car going around a particular banked turn. The centripetal force needed to turn the car (mv²/r) depends on the speed of the car since the mass of the car and the radius of the turn are fixed in the expression mv²/r  If the car is at high speed, it requires more centripetal force and if it is at less speed requires less centripetal force.

The forces acting on the body are the Normal force N and the frictional force f, as shown in the diagram.

The forces N and f can be resolved into horizontal and vertical components. We can balance the horizontal components and vertical components separately.

Thus, In the vertical direction (Y axis)

NcosƟ = fsinƟ + mg --------------------(i)

In horizontal direction (X axis)

fcosƟ + NsinƟ = mv²/r  ----------------(ii)

Since we know that f =  μ_sN   where μ_s is the co-efficient of friction

For maximum velocity, f = μ_sN. Hence, (i)becomes:

NcosƟ = μ_sNsinƟ + mg

Or, NcosƟ - μ_sNsinƟ = mg

Or, N = mg/(cosƟ- μ_ssinƟ)

Put the above value of N in (ii)

μ_sNcosƟ + NsinƟ = mv²/r 

μ_smgcosƟ/(cosƟ- μssinƟ) + mgsinƟ/(cosϴ- μssinƟ) = mv²/r 

mg (sinƟ + μ_scosƟ)/ (cosƟ - μ_ssinƟ) = mv²/r 

Divide the Numerator & Denominator by cosƟ, we get

v2 = Rg (tanƟ +μ_s) /(1- μ_s tanƟ)

v = √ Rg (tanƟ +μ_s) /(1- μ_s tanƟ)

This is the maximum speed of a car on a banked road when friction is also present.