Bicycle & E-Bike Power, Gearing and Range Calculator

The steady road-load model follows the force-balance approach of Martin et al. (1998), DOI 10.1123/jab.14.3.276 (https://doi.org/10.1123/jab.14.3.276). The attached paper's worked example is on file page 16 (printed p.291). This implementation retains gravity, rolling resistance and longitudinal drag; it omits the paper's bearing and acceleration terms.

For grade fraction $s$, $\theta=\arctan(s)$. With ground speed $v$ and signed headwind $w$:

Wheel demand is $P_w=\max[0,(F_g+F_r+F_a)v]$. Negative road load is reported as coasting braking demand. Required unassisted crank power is $P_w/\eta_c$. Rider power is a budget and is reduced when demand is smaller.

Gearing is derived from no-slip chain and wheel kinematics: wheel/crank speed ratio is front/rear teeth; rollout is $\pi D$ times that ratio; gear inches are rolling diameter in inches times the ratio. Angular speed is explicitly converted between rad/s and rpm. Up to nine distinct grid rows are tooth-count candidates, not catalogue data. The row count is limited by the number of integer tooth counts between the chosen bounds.

The motor supplies demand left after the rider. Its output envelope is the smaller of mechanical power and output torque times angular speed, multiplied by assistance permission. A mid-drive loses chain power; a hub acts at the wheel. EU assistance is assumed to taper linearly over the last 1 km/h below cutoff; this taper width is an implementation assumption. Other presets use an ideal sharp cutoff. A requested speed with more than 0.5 W of wheel deficit is flagged and has no range result.

Battery draw is motor output divided by efficiency plus accessory draw. Wh/km equals battery watts divided by ground speed in km/h. Usable Wh includes health, the starting-charge minus reserve window, and the cold factor $\mathrm{clamp}[1-\alpha\max(20-T,0),0.2,1]$. Range is usable Wh divided by Wh/km. Zero draw has no finite battery-limited range.

Heating is a derived first-order energy balance: $\Delta T(t)=P_{loss}R_{th}(1-e^{-t/(R_{th}C_{th})})$. The thermal node starts at ambient. All drive conversion loss is assigned to it; chain and accessory losses are excluded. The requested segment time is kinematic. Heating is withheld for an infeasible speed or a pedal-only bicycle. For a feasible assisted case, the powered interval ends at segment completion or battery reserve, whichever comes first: $t_h=\min(t_{segment},E_{usable}/P_{battery})$. With zero battery draw it ends at segment completion. The temperature card is the estimate at this interval's end; it does not predict completion of a longer, battery-depleted segment or behavior after reserve.

In the scene, the drive arrow represents actual rider plus motor traction. Air force acts opposite the signed aerodynamic resistance; a faster tailwind therefore points forward. Coasting braking appears when road load is negative. These longitudinal arrows are aligned with the illustrated road; weight remains vertical.

• One uniform road segment, steady speed, constant rider budget and no-slip rolling. Mass includes the battery and luggage. No pedalling physiology model is used.
• Default drag area, rolling coefficient, efficiencies, torque limit, cold slope and thermal parameters are editable scenario assumptions, not a measured bicycle or a product specification.
• Air density is entered independently. Ambient temperature is also the assumed battery temperature; a warm indoor-stored battery can behave differently.
• The cold slope defaults to 0.01 K⁻¹: 80% retained capacity at 0 °C and 60% at −20 °C, before health and reserve. Bosch's winter guidance supports reduced capacity qualitatively, not these numerical factors.
• Motor torque is at the crank or wheel after internal reduction. Fixed efficiency omits speed/load-dependent copper, iron and controller loss. The one-node thermal estimate is not a winding hotspot prediction.
• Crank cadence is a kinematic value; a hub or throttle-driven bicycle can travel while its rider coasts. The rider animation pedals only when the rider supplies positive power. Bicycle position is fixed, and road angle is exaggerated fourfold then capped at −20° and +24° for legibility. Arrow directions and visibility follow calculated forces; arrow lengths are schematic.
• Physics equations for gearing, energy accounting and lumped heating are derived here. Published cases independently check road-load components, simplified cycling examples and a gear-chart entry.

Use for preliminary bicycle comparison and teaching, not motorcycle homologation or a guaranteed trip range. Starts, stop-start traffic, crosswind/yaw, drafting, tire slip, regenerative braking, battery voltage sag, BMS limits, motor speed ceilings and route profiles are excluded. Descents report the braking power needed to hold the chosen speed; they do not certify brake or tire safety. Thermal parameters need measurements at the relevant airflow and load.

Regulatory preset date: 26 October 2026. EU: Regulation (EU) 168/2013, Article 2(2)(h), pedelec exemption: maximum continuous rated power ≤250 W, pedalling required, assistance progressively reduced and cut off by 25 km/h. US: a 750 W, 20/20/28 mph three-class planning model, illustrated by Indiana DNR's definitions; it is not a nationwide legal certification. Class 1/3 require pedalling here; Class 2 allows motor-only operation. State, local, age and access rules must be checked separately.

China: GB 17761-2024, published 31 December 2024 and effective 1 September 2025; this preset uses 400 W and 25 km/h design/assistance limits. MIIT distinguishes motor-driven vehicles from pedal-assisted vehicles and explains that the standard does not require pedals on all motor-driven types. The national non-motor-lane operating speed rule is 15 km/h, distinct from the 25 km/h design limit. These presets do not check construction, mass, electrical safety, registration or certification. A legal continuous rating is not a universal instantaneous peak-power limit; the calculator deliberately uses a simple planning ceiling. Check against the standard in force and the rules at the place of use.