Max Turning Velocity on a Banked Curve

The Maximum Velocity of a Car on a Banked Curve calculator computes the maximum velocity that a car can go on a banked curve where the centrifugal force outward and upward does not overcome both the downward force of gravity and the force of friction of the tires on the surface.

INSTRUCTIONS: Choose units and enter the following:

• (r) Radius of Curve
• (Φ) Angle of the Curve
• (μ) Coefficient of Friction

Max Velocity (Vmax): The calculator returns the velocity in kilometers per hour.  However this can be automatically converted to compatible units via the pull-down menu. 

The Math / Science

The equation for the maximum velocity a car could achieve on a banked surface is:

`v_(max) = sqrt(    (#r*g * (sin(#phi) + #mu * cos(#phi))) / (cos(#phi) - #mu  * sin(#phi))    )`

where:

• v_max = max velocity before the outward force exceed traction and gravity.
• r = the radius of the curve
• `phi` = the angle of the banked surface
• `mu` = the coefficient of friction characterizing the two surfaces interaction.

In the illustration, we see the back end of a car turning to the left.  The weight of the car due to the force of gravity pushes downward.  That force is equal and opposite to the vertical component of the force of the road pushing up on the car's tires.  The frictional force, `F_"Friction"`, is toward the direction of the turn and parallel to the banked surface and keeps the car from sliding outward during the turn. 

The force of the road pushing up (perpendicular to the road surface) on the car is is the force `F_"Road"`.

We can determine the maximum velocity, v,  by assuming the horizontal forces sum exactly to zero.  We first compute the force exerted on the car's tires by the road, `F_"Road"`:

Eq. 1: `sum F_"Vertical" = F_"Road" * cos(phi) - F_"Road" * mu * sin(phi) - m*g = 0`

Thus we can compute `F_"Road" * mu = F_"Friction"` from Eq. 1 as:

Eq. 2: `F_"Road" * (cos(phi) - mu  * sin(phi)) =  mg`

Eq. 3: `F_"Road" = mg`/`(cos(phi) - mu  * sin(phi))`

We then substitute `F_"Road"` into the following equation:

Eq. 4: `sum F_"Horizontal" = F_"Road" * sin(phi) + mu*F_"Road" * cos(phi) = m*v^2/r`

which provides us:

Eq. 5: `((m*g)/ (cos(phi) - mu  * sin(phi)))  * sin(phi) + ((m*g)/(cos(phi) - mu  * sin(phi))) * mu * cos(phi) =m*v^2/r`

Eq. 6: `((r*g) / (cos(phi) - mu  * sin(phi)))  *  (sin(phi) + mu * cos(phi)) = v^2`

Eq. 7: `v = sqrt(    (r*g * (sin(phi) + mu * cos(phi))) / (cos(phi) - mu  * sin(phi))    )`