Banked Curve
Ideal banking angle, and the band of safe speeds once friction is included.
33.47 km/h — needs no friction at all
79.71 km/h
the bank is shallow enough that it cannot slide down
| Ideal banking angle for that speed | 10° | tan θ = v² ÷ (rg) — the same angle you entered, by construction |
| Safe speed band | 0 to 22.14 m/s | friction widens the single ideal speed into a range |
| Normal force at the ideal speed | 11949.52 N | mg ÷ cos θ — larger than the weight, since the road also turns the car |
| Centripetal force needed | 2075.01 N | mv² ÷ r, supplied entirely by the horizontal part of the normal force |
| Weight | 11767.98 N | mg with g = 9.80665 m/s² |
Mass cancels out of every speed here — a loaded lorry and an empty car have the same ideal speed on the same bend. It only changes the forces involved.
Banking a bend lets the road itself turn the car. Tilt it at the right angle and no friction is needed at all: tan θ = v² ÷ (rg). Friction then widens that single ideal speed into a range you can safely drive.
Why tilting the road helps
Turning needs a force pointing at the centre of the bend. On a flat road only friction can supply it, so the grip available sets the maximum speed, and on ice there is almost nothing to work with. Tilt the road and the normal force — which is always perpendicular to the surface — acquires a horizontal component pointing inward. The road starts doing the turning.
At one particular speed that horizontal component is exactly what the turn requires, and friction contributes nothing. Setting N sin θ equal to mv²/r and N cos θ equal to mg, then dividing one by the other, gives tan θ = v² ÷ (rg). The normal force cancels and so does the mass, which is why the ideal speed for a bend is the same for a motorcycle and a loaded lorry.
Friction turns a speed into a band
Real drivers do not all travel at the ideal speed, and friction takes up the difference. Go faster and the car tends to slide up the bank, so friction acts down it; go slower and it tends to slide down, so friction acts up. Each case gives a limit, and between them lies a range of safe speeds rather than a single value.
v<sub>min</sub> swaps the signs: √( rg · (tan θ − μ) ÷ (1 + μ tan θ) )
- 1 Draw the three forces. Weight straight down, normal force perpendicular to the road surface, friction along it.
- 2 Resolve horizontally and vertically, not along the slope. The acceleration is horizontal — toward the centre of the bend — so those are the useful directions. Resolving along the incline, as for a block on a ramp, makes this harder than it needs to be.
- 3 Set the horizontal resultant equal to mv² ÷ r. Vertically the forces balance, because the car is not moving up or down.
- 4 Divide the two equations. Both N and m cancel, leaving tan θ = v² ÷ (rg) in the frictionless case.
- 5 Add friction for the limits. At each limit friction is fully used, at μN — pointing down the bank at the maximum speed and up it at the minimum.
Ideal speed for a given bend
The speed at which no friction is needed. Independent of the vehicle’s mass.
| Radius | 5° | 10° | 25° | 45° |
|---|---|---|---|---|
| 25 m | 4.6 m/s | 6.6 m/s | 10.7 m/s | 15.7 m/s |
| 50 m | 6.5 m/s | 9.3 m/s | 15.1 m/s | 22.1 m/s |
| 100 m | 9.3 m/s | 13.1 m/s | 21.4 m/s | 31.3 m/s |
| 200 m | 13.1 m/s | 18.6 m/s | 30.2 m/s | 44.3 m/s |
What the model leaves out
The first thing to notice is what it says about mass: nothing. Every speed here is independent of it, because the weight that has to be supported and the centripetal force that has to be supplied both scale with mass in the same way. Only the forces themselves change, which is why the tool reports those separately.
The second is the case where the mathematics stops being useful. When μ × tan θ reaches 1 the denominator in the maximum-speed formula vanishes and the model reports no upper limit at all. That is a genuine property of the equations — on a steep enough bank with enough grip, friction grows faster than the force required — but it is not a statement about any real road. Tyre limits, the camber of the vehicle and the risk of rolling rather than sliding all bite long before this point.
Two more simplifications worth naming. The car is treated as a point mass, so rolling over is never considered, though for a tall vehicle that often happens before sliding does. And μ is static friction, which is what applies while the tyre is not slipping — once it does slip the coefficient drops and the limits tighten, which is why a skid is so hard to recover from.