Physics · Classical Mechanics · Dynamics & Forces
Cycling Power to Speed Calculator
Calculate cycling speed from power output accounting for aerodynamic drag, rolling resistance, gradient, and rider mass.
Calculator
Formula
P is power in watts; rho (ρ) is air density (kg/m³); Cd is drag coefficient; A is frontal area (m²); v is speed (m/s); Crr is rolling resistance coefficient; m is total mass (kg); g is gravitational acceleration (9.81 m/s²); theta (θ) is road gradient angle (radians from percent grade); Fbrake is any braking force (typically 0 for steady-state).
Source: Martin, J.C. et al. (1998). Validation of a Mathematical Model for Road Cycling Power. Journal of Applied Biomechanics, 14(3), 276–291.
How it works
The fundamental equation of cycling power balances the mechanical work done per second against three resistive forces: aerodynamic drag, rolling resistance, and gravitational climbing force. Aerodynamic drag power scales with the cube of velocity and is proportional to air density (ρ), the drag area CdA, and speed cubed: P_aero = 0.5 × ρ × CdA × v³. Rolling resistance power is linear in speed: P_rr = Crr × m × g × cos(θ) × v. Gravitational power on a slope is: P_climb = m × g × sin(θ) × v, where θ is derived from the percentage grade.
Because the total power equation is a cubic polynomial in speed, there is no simple algebraic closed form for v. This calculator solves for speed numerically using a bisection method, iterating 200 times to converge to a precise answer. Drivetrain efficiency is applied by multiplying the input power by (1 − loss%), reflecting that a chain drive typically delivers 97–99% of pedalled power to the rear wheel.
The model is widely used in competitive cycling and sports science. Understanding how power splits between aero drag, rolling resistance, and climbing helps riders make evidence-based decisions about position, tyres, body weight, and altitude. The output breakdown shows exactly how many watts are consumed by each resistance component at the calculated speed.
Worked example
Example: A rider produces 250 W with a combined rider+bike mass of 80 kg, on a flat road (0% gradient), CdA of 0.32 m², Crr of 0.004, air density of 1.225 kg/m³, and 2% drivetrain loss.
Step 1 — Effective power: 250 × (1 − 0.02) = 245 W
Step 2 — Resistance coefficients: A = 0.5 × 1.225 × 0.32 = 0.196; B = 0.004 × 80 × 9.81 × 1.0 = 3.139; C = 0 (flat road).
Step 3 — Bisection solve: We need v such that 0.196v³ + 3.139v = 245. Testing v = 9 m/s: 0.196×729 + 3.139×9 = 142.9 + 28.3 = 171.2 W (too low). Testing v = 10 m/s: 0.196×1000 + 3.139×10 = 196 + 31.4 = 227.4 W (too low). Testing v = 11 m/s: 0.196×1331 + 3.139×11 = 260.9 + 34.5 = 295.4 W (too high). Converging between 10 and 11 m/s, the bisection yields approximately 10.6 m/s = 38.2 km/h.
Step 4 — Power breakdown at 10.6 m/s: Aero drag = 0.196 × 10.6³ ≈ 233 W; Rolling = 3.139 × 10.6 ≈ 33 W. Aero drag accounts for ~87% of resistance at this speed — a typical flat-road result.
Limitations & notes
This model assumes steady-state speed with no acceleration, and that the rider travels in still air (no headwind or tailwind). In reality, wind speed and direction dramatically affect aerodynamic drag — a 20 km/h headwind can double the effective drag at typical cycling speeds. The model also assumes a fixed CdA; in practice, frontal area changes with rider position and fatigue. Rolling resistance is treated as constant, but in reality it varies with tyre pressure, road surface, and cornering. The bisection solver assumes the power function is monotonically increasing in speed, which holds for all physically plausible inputs. Inputs with negative net power at any speed (e.g. extreme downhill) may produce unexpectedly large speed values — always apply physical judgement to outputs.
Frequently asked questions
What is CdA and how do I find my value?
CdA (drag coefficient × frontal area, in m²) is the single most important aerodynamic parameter in cycling. Typical values range from about 0.20 m² for a professional in a full TT tuck to 0.40 m² or more for an upright recreational rider. You can estimate your CdA from a velodrome test, a field test on a calm flat road (using power and speed data), or a wind tunnel session. Many online resources list typical values by riding position: road drops ~0.32 m², TT position ~0.24 m², upright bars ~0.45 m².
Why does aerodynamic drag dominate at high speeds?
Aerodynamic drag power scales with the cube of speed (v³), while rolling resistance power scales linearly (v). At low speeds (below ~15 km/h) rolling resistance is the dominant force, but as speed increases the cubic term grows rapidly. By 40 km/h on a flat road, aerodynamic drag typically accounts for 80–90% of all resistive power. This is why professional time trialists invest heavily in aerodynamic equipment and position rather than in tyre rolling resistance alone.
What air density should I use?
Air density depends on altitude, temperature, and humidity. At sea level on a standard day (15°C, 1013 hPa), ρ = 1.225 kg/m³. At altitude — for example, 2000 m elevation — density drops to about 1.006 kg/m³, which significantly reduces aerodynamic drag and explains why many hour-record attempts are made at altitude. You can calculate air density from your local weather data using the ideal gas law, or use 1.225 kg/m³ as a reasonable sea-level default.
What is a typical rolling resistance coefficient (Crr) for road cycling?
Crr for modern road cycling tyres on asphalt typically ranges from 0.003 to 0.006. High-end latex-tube continental tyres can achieve Crr as low as 0.003; butyl-tube training tyres are typically 0.004–0.005; and wider mountain bike tyres may reach 0.010–0.020. Lower tyre pressure increases Crr. The difference between a Crr of 0.003 and 0.005 at 40 km/h on a flat road equates to roughly 10–15 W, which is meaningful at elite level but modest compared to aerodynamic differences.
How does road gradient affect the power required?
On a climb, every watt invested against gravity is proportional to mass and grade. At 5% gradient with an 80 kg rider+bike, climbing power at 20 km/h is approximately 80 × 9.81 × sin(atan(0.05)) × 5.56 ≈ 218 W, which already exceeds the aero drag component. This explains why lighter riders have a significant advantage in the mountains and why the power-to-weight ratio (W/kg) is the key metric for climbing performance, while flat-road performance is governed more by absolute power and aerodynamics.
What drivetrain loss should I enter?
A well-maintained, properly lubricated chain drivetrain typically loses 1–2% of pedalled power. A dirty or poorly lubricated drivetrain can lose 3–5%. Single-speed or belt-drive systems may be slightly more efficient, while some gear combinations (e.g. cross-chaining on a derailleur) can increase losses. For most practical calculations, 2% (enter 2) is a reasonable default that reflects a clean road bike drivetrain.
Last updated: 2025-01-30 · Formula verified against primary sources.