E-Bike Range Calculator
E-Bike Range Calculator
How far an e-bike battery goes, built from the physics rather than a lookup: rolling resistance, aerodynamic drag and gradient worked out separately, divided by the drive efficiency, and turned into watt-hours per kilometre and a range — with the split shown, because at 25 km/h the air is taking most of it.
e-bike range
a 36 V 14 Ah pack used to 80%, 95 kg all up, averaging 25 km/h on the flat, Cᵣᵣ 0.0077, CdA 0.559 m², drive 80% efficient
The three forces, and what they cost
θ = arctan(gradient ÷ 100) Pwheel = F v
Wh/km = (Pwheel − Prider) ÷ η ÷ speed(km/h)
range = usable Wh ÷ (Wh/km) usable Wh = V × Ah × DoD
- C rr
- rolling resistance coefficient: the fraction of the normal load that appears as a retarding force. Almost independent of speed, so it costs the same per kilometre however fast you go
- Cd A
- drag area — the drag coefficient times the frontal area, quoted as one number because only the product is measurable. Force goes with v², power with v³
- rho
- air density, 1.225 kg/m³ at sea level and 15 °C
- theta
- road angle. A gradient in per cent is a tangent, so θ is its arctangent — using the percentage directly as a sine is 2% wrong at a 20% grade
- eta
- battery-to-wheel efficiency: controller, motor and transmission together, typically 0.75 to 0.85
Worked example
a 36 V 14 Ah pack used to 80%, 95 kg all up, averaging 25 km/h on the flat, Cᵣᵣ 0.0077, CdA 0.559 m², drive 80% efficient
Usable energy = 36 × 14 × 80% = 403.2 Wh out of 504.0 Wh in the pack
Rolling: Cᵣᵣ m g = 0.0077 × 95 × 9.80665 = 7.174 N, and it is the same at any speed
Drag at 25 km/h (6.9444 m/s): ½ρCdA v² = 0.5 × 1.225 × 0.559 × 48.225 = 16.512 N — already 69.7% of the total
Total force 23.685 N, so the wheel needs 23.685 × 6.9444 = 164.5 W, and at 80% efficiency the battery supplies 205.6 W — 5.71 A
Energy per kilometre = 205.6 W ÷ 25 km/h = 8.224 Wh/km
Range = 403.2 ÷ 8.224 = 49.0 km, about 1.96 hours of riding
Drag overtakes rolling resistance at 16.5 km/h, so above walking pace the air is the main thing the battery is paying for
Where the energy goes, at speeds people actually ride
| Speed | Rolling | Drag | Total force | Power at the wheel | Wh/km | Range | Drag’s share |
|---|---|---|---|---|---|---|---|
| 10 km/h | 7.17 N | 2.64 N | 9.82 N | 27 W | 3.41 | 118.3 km | 27% |
| 15 km/h | 7.17 N | 5.94 N | 13.12 N | 55 W | 4.55 | 88.5 km | 45% |
| 20 km/h | 7.17 N | 10.57 N | 17.74 N | 99 W | 6.16 | 65.5 km | 60% |
| 25 km/h | 7.17 N | 16.51 N | 23.69 N | 164 W | 8.22 | 49.0 km | 70% |
| 30 km/h | 7.17 N | 23.78 N | 30.95 N | 258 W | 10.75 | 37.5 km | 77% |
| 35 km/h | 7.17 N | 32.36 N | 39.54 N | 384 W | 13.73 | 29.4 km | 82% |
| 40 km/h | 7.17 N | 42.27 N | 49.44 N | 549 W | 17.17 | 23.5 km | 85% |
Range is physics, not a number on a box
Manufacturers quote e-bike range as a single figure, usually measured by a light rider on flat ground at low assist with a fresh battery. It is not a lie exactly, but it answers a question nobody asked. The honest answer is built from three forces, each of which behaves differently, and this page computes them separately so you can see which one you are actually paying for.
Rolling resistance is the tyre and the road: a retarding force of C_rr × m × g × cos θ, very nearly independent of speed. That means it costs the same energy per kilometre whether you ride at 15 km/h or 40 — it is a distance tax, not a speed tax. It scales straight with mass, which is the one place a heavier rider or a loaded pannier really shows up. The default here, 0.0077, is the measured mean across 557 urban cyclists in Vancouver; high-pressure road tyres reach 0.004 and soft wide tyres exceed 0.012, so the honest range of this one number is a factor of three.
Aerodynamic drag is ½ρ·CdA·v². Force goes with the square of speed and the power it demands with the cube, so it starts out negligible and then takes over completely. On the default bike the crossover — where drag first exceeds rolling resistance — is at about 16 km/h. By 25 km/h, which is where European pedelec assist cuts out, drag is already about 70% of the total force. That is the single most useful thing on this page: at any speed you would actually call cycling, most of the battery is being spent on air. It follows that riding 5 km/h slower does more for range than any plausible weight saving, and that a headwind is brutal, because what matters is airspeed, not ground speed.
Gradient is m·g·sin θ, and on any real hill it swamps the other two. It is also the one term that is pure physics with no coefficient to argue about — lifting 95 kg by 100 m takes 93 kJ, or 26 Wh, whatever the bike. One detail worth getting right: a gradient quoted as a percentage is a tangent, rise over run, so the angle is its arctangent and the useful terms are sin and cos of that. Using the percentage directly as a sine, which many calculators do, is 0.5% wrong at 10% and 2% wrong at 20%.
And then the two honest fudges. Drive efficiency covers the controller, the motor and the transmission; 75 to 85% is realistic for a hub motor at a sensible load and it falls away badly at low speed or on a steep climb in the wrong gear, which is exactly when you need it most. Depth of discharge covers the battery management system’s cut-off and the fact that repeatedly running a lithium pack flat shortens its life; 80% is a fair working figure. Both are inputs here rather than hidden constants, because they are the two places a range estimate is most often quietly optimistic.
What this model leaves out. Acceleration: stop-start city riding throws away the kinetic energy at every junction, and this page assumes a steady speed. Wind: only airspeed matters to the drag term, and a 15 km/h headwind at 25 km/h ground speed more than doubles the drag force. Cold: a lithium pack at 0 °C delivers roughly 20 to 30% less usable energy, which returns when it warms. Battery ageing, which takes a few per cent a year. And regeneration, which most e-bikes do not have and which recovers only a small fraction on the ones that do. For converting the pack’s rating between amp-hours and watt-hours, see the mAh to Wh converter; for how long the pack takes to fill again, the battery charging time calculator; and for the motor’s own torque and speed at the shaft, the motor power, torque and speed calculator.
Frequently asked questions
How do I calculate e-bike range?
Usable watt-hours divided by watt-hours per kilometre. The usable energy is volts × amp-hours × the share of the pack you will really use; the watt-hours per kilometre come from the resisting force divided by the drive efficiency. On a 36 V 14 Ah pack at 80% depth of discharge, a 95 kg rider and bike at 25 km/h on the flat uses about 8.2 Wh/km and gets about 49 km.
Why does riding faster use so much more battery?
Because aerodynamic drag rises with the square of speed, and the power it takes with the cube. At 15 km/h drag is a minority of the total force; at 25 km/h it is about 70%; at 35 km/h it is nearly 85%. Slowing down by 5 km/h is the cheapest range you will ever buy.
Does a heavier rider reduce e-bike range?
On the flat, less than people expect: mass only enters the rolling resistance term, which is a minority of the total at normal speeds. Ten extra kilograms on the default bike adds about 0.75 N of rolling resistance out of 24 N. On hills it is completely different — the gradient term is proportional to mass and dominates everything else.
How much range do I lose in winter?
Roughly 20 to 30% at freezing, most of it from the battery rather than the bike: a cold lithium pack cannot deliver its full energy. It comes back when the pack warms up, so the capacity is not lost permanently. Denser cold air adds a few per cent of drag on top.
What is a realistic Wh/km for an e-bike?
Between about 5 and 15 Wh/km for most riding. Light assist on the flat at 15 km/h can be under 5; hard riding at 30 km/h into a headwind, or a hilly route with a loaded cargo bike, easily exceeds 20. This page works out yours instead of guessing.
Does pedalling really extend the range?
Directly and proportionally. Every watt you contribute is a watt the battery does not supply, divided by the drive efficiency — so 75 W of steady pedalling on the default bike is nearly 94 W off the battery, close to half the demand. That is the whole idea of a pedelec.
Related calculators
References
- Tengattini S, Bigazzi AY. Physical characteristics and resistance parameters of typical urban cyclists. Journal of Sports Sciences, vol. 36 no. 20, 2018, pp. 2383–2391. Field measurements on 557 cyclists in Vancouver: mean rolling resistance coefficient 0.0077 (SD 0.0036), mean effective frontal area 0.559 m² (SD 0.170), mean bicycle-plus-cargo mass 18.3 kg (SD 4.1). Those three numbers are this page’s defaults.
- Wilson DG, Papadopoulos J. Bicycling Science, 3rd ed. MIT Press, 2004. Chapters 5 and 6 — the resistance model P = (C_rr m g + ½ρC_dA v² + m g sin θ)v, roller measurements giving rolling coefficients of 0.003 to 0.006 for well inflated tyres, and the air density of 1.225 kg/m³ at sea level and 15 °C.
- ISO 2533:1975, Standard Atmosphere. Sea-level density 1.225 kg/m³ at 15 °C and 101.325 kPa — the default used here, and the source of the roughly 10% fall per 1,000 m of altitude mentioned above.
- Regulation (EU) No 168/2013, Article 2(2)(h) — the exclusion for pedal cycles with pedal assistance whose motor output is cut off at 25 km/h and whose continuous rated power does not exceed 250 W. That 25 km/h is why the default speed here is what it is.
- BIPM. The International System of Units (SI), 9th ed., 2019. Standard acceleration of gravity g = 9.806 65 m/s² exactly, used for both the rolling and the gradient terms.