Motion of a car on a banked road

               Consider about a car of mass m kg moves on banked road of inclination θ. During the motion the car experience circular motion.

Mathematical Calculation

There is no acceleration along the vertical direction. The net force along this direction must be zero.

N cosθ = mg + f sinθ ------------ (i)

The horizontal components of N and f provide the centripetal force.

N cosθ + f cosθ = mv²/r ---------- (ii)

The frictional force 

f ≤ μ_s N
For maximum velocity v

f = μ_s N
Now putting the value of f in equation no. (i)

N cosθ = mg +  μ_s N sinθ
Therefore,
N = mg/(cosθ  - μ_s sinθ)

Again putting the value of f in equation no. (ii)

N sinθ + μ_s N cosθ = mv²/r

Now substitute the value of N

mg( sinθ  + μ_s  cosθ)/(cosθ  - μ_s sinθ) = mv²/r

Therefore,

v = {rg( sinθ  + μ_s  cosθ)/(cosθ  - μ_s sinθ)}^1/2

This is maximum velocity of a car on a banked road.