A banked roadway is a circular road tilted by an angle \(\theta\). The tilt helps provide the inward force needed for
circular motion. On a flat road, the inward force must come entirely from static friction. On a banked road, the normal
force is tilted, so it has an inward horizontal component.
Frictionless design speed
At the ideal design speed, no static friction is required. The only contact force is the normal force \(N\), tilted inward
by the bank angle. The vertical component balances weight, and the horizontal component supplies centripetal force.
Force balance without friction.
\[
\begin{aligned}
N\cos\theta &= mg \\
N\sin\theta &= \frac{mv_0^2}{r}
\end{aligned}
\]
Dividing the horizontal equation by the vertical equation eliminates both \(N\) and \(m\):
Design speed formula.
\[
\begin{aligned}
\tan\theta &= \frac{v_0^2}{rg} \\
v_0 &= \sqrt{rg\tan\theta}
\end{aligned}
\]
This speed is called the design speed because a car traveling at this speed needs no sideways friction to stay on the
circular path.
Why friction gives a speed range
With static friction available, a banked road can support speeds below and above the frictionless design speed. At low speed,
the car tends to slide down the bank, so static friction acts up the bank. At high speed, the car tends to slide up the bank,
so static friction acts down the bank.
Low-speed limiting case: friction up the bank.
\[
\begin{aligned}
\frac{mv_{\min}^{2}}{r} &= N\sin\theta-\mu_sN\cos\theta \\
mg &= N\cos\theta+\mu_sN\sin\theta
\end{aligned}
\]
Solving these equations gives:
Minimum safe speed.
\[
\begin{aligned}
v_{\min} &= \sqrt{rg\frac{\sin\theta-\mu_s\cos\theta}{\cos\theta+\mu_s\sin\theta}}
\end{aligned}
\]
If the numerator \(\sin\theta-\mu_s\cos\theta\) is zero or negative, then static friction is strong enough to hold the car
even at rest, so the minimum speed is \(0\).
High-speed limiting case: friction down the bank.
\[
\begin{aligned}
\frac{mv_{\max}^{2}}{r} &= N\sin\theta+\mu_sN\cos\theta \\
mg &= N\cos\theta-\mu_sN\sin\theta
\end{aligned}
\]
Solving the high-speed limiting equations gives:
Maximum safe speed.
\[
\begin{aligned}
v_{\max} &= \sqrt{rg\frac{\sin\theta+\mu_s\cos\theta}{\cos\theta-\mu_s\sin\theta}}
\end{aligned}
\]
If \(\cos\theta-\mu_s\sin\theta\le 0\), the simplified model predicts no finite upper-speed limit. In practice, real tires,
road surfaces, suspension, and safety constraints impose additional limits.
Mass cancellation
The speed formulas do not contain mass. This happens because weight, normal force, friction, and required centripetal force
all scale with the car’s mass. A heavier car requires more centripetal force, but it also has proportionally larger weight,
normal force, and friction capacity in this idealized model.
Mass is still useful if you want actual force magnitudes. At the design speed, the normal force and centripetal force are:
Design-speed forces.
\[
\begin{aligned}
N_0 &= \frac{mg}{\cos\theta} \\
F_{c,0} &= m\frac{v_0^2}{r}
\end{aligned}
\]
Summary of formulas
| Case |
Formula |
Meaning |
| Frictionless design |
\(v_0=\sqrt{rg\tan\theta}\) |
Speed where no static friction is needed |
| Low-speed limit |
\(v_{\min}=\sqrt{rg(\sin\theta-\mu_s\cos\theta)/(\cos\theta+\mu_s\sin\theta)}\) |
Friction acts up the bank |
| High-speed limit |
\(v_{\max}=\sqrt{rg(\sin\theta+\mu_s\cos\theta)/(\cos\theta-\mu_s\sin\theta)}\) |
Friction acts down the bank |
| Design acceleration |
\(a_{c,0}=v_0^2/r=g\tan\theta\) |
Inward centripetal acceleration at design speed |
This model assumes a rigid car, a uniform bank angle, a circular path, constant static friction coefficient, and no aerodynamic
downforce or suspension effects.
