Cycling & Mountain Biking · Physics of Power, Drag & Grip

02 · The science

Six chapters. Every number worked out.

Each chapter explains one idea in plain language, then works a real example. Try the Power Lab to see where your watts go on the flat, uphill and off-road.

Interactive · Power Lab

Set your watts, the slope, the surface and your position, and see where every watt goes

A simple model: your power is shared between rolling resistance (C_rr × weight), climbing (weight × sin θ) and air drag (½ρC_dAv², with sea-level air and no wind). It ignores drivetrain losses, so real speeds are a little lower. Drag areas and rolling coefficients are illustrative. For learning only.

Your power200 W

Gradient0%

Rider + bike mass85 kg

Position

Surface

Steady speed33.9 km/hCh 1 · power = force × speed

Watts per kilo2.35 W/kgCh 3 · what climbing rewards

Power to air drag80%Ch 2 · grows with v³

Power to climbing0%Ch 3 · m g sin θ × v

Power to rolling20%Ch 6 · tyres and surface

Time for 10 km17.7 minat this steady speed

Height gained per hour0 m/hCh 3 · climbing rate

Bar = speed on a 0–50 km/h scale. Try 8% with each mass, then flat with each position, then the dirt trail. Ride at speeds that suit the road, the traffic and your brakes (Ch 5).

CH 1

Power & the three resistances

Every watt you make goes into rolling, climbing or pushing air.

Power is force × speed: P = F × v. On a bike, three forces resist you: rolling resistance (C_rr × m g, from tyres squashing), gravity on a slope (m g sin θ), and air drag (½ ρ C_dA v²). Rolling and gravity grow in step with speed in power terms; drag power grows with v³.

Worked example · 85 kg rider + bike, flat smooth road, C_rr 0.005, C_dA 0.32 m², ρ 1.2 kg/m³

Speed	Rolling power	Drag power	Total
20 km/h	≈ 23 W	≈ 33 W	≈ 56 W
30 km/h	≈ 35 W	≈ 111 W	≈ 146 W
40 km/h	≈ 46 W	≈ 263 W	≈ 309 W

At 30 km/h (8.33 m/s): rolling = 0.005 × 85 × 9.81 × 8.33 ≈ 35 W; drag = ½ × 1.2 × 0.32 × 8.33³ ≈ 111 W. Doubling speed from 20 to 40 km/h needs more than five times the power. Drag areas and C_rr are typical illustrative values.

✋

Feel the numberA bright household LED bulb uses about 10 W. Cruising at 30 km/h on the flat takes about 150 W: fifteen of them, from your legs, for as long as you ride.

You hold 300 W against 30 N of total resistance. How fast are you going?

v = P ÷ F = 300 ÷ 30 = 10 m/s = 36 km/h.

↑ Back to the challenge

CH 2

Air drag, position & drafting

Your body, not the bike, makes most of the drag. How you sit changes everything.

The drag area C_dA combines how big you look from the front and how streamlined you are. Sitting upright on a mountain bike, you're a parachute; low on the drops of a road bike, you slice through. The same power then gives very different speeds.

Worked example · 200 W, 85 kg, flat smooth road (from our Power Lab)

Position	C_dA (illustrative)	Steady speed	10 km takes
Upright (MTB)	0.60 m²	≈ 27.9 km/h	≈ 21.5 min
Hands on the hoods	0.32 m²	≈ 33.9 km/h	≈ 17.7 min
Low, on the drops	0.27 m²	≈ 35.7 km/h	≈ 16.8 min

Solve 200 = v(0.005 × 85 × 9.81) + ½ × 1.2 × C_dA × v³ for v. With C_dA 0.32: v ≈ 9.42 m/s ≈ 33.9 km/h; about 80% of the 200 W goes into drag.

Drafting

A rider ahead pushes the air aside and leaves a wake of slower-moving air. Ride in it and your drag falls, which is why groups ride in lines and take turns at the front. The cost: you're centimetres from another wheel, with little time to react. Ride close only with riders you trust, never on a busy road, and always signal hazards.

✋

Feel the numberSitting up into a headwind feels like hitting a wall because drag depends on your speed through the air, not over the ground. A 15 km/h headwind at 25 km/h feels like riding at 40 km/h in still air.

If you go 10% faster, by roughly how much does your drag power rise?

1.1³ ≈ 1.33: about a third more power, for a tenth more speed.

↑ Back to the challenge

CH 3

Climbing & watts per kilo

Uphill, the air hardly matters. Your weight does.

On a gradient, gravity pulls back with m g sin θ, where tan θ = gradient ÷ 100. Climbing is slow, so drag is small and most of your power lifts you and the bike. That's why climbers talk about watts per kilo and height gained per hour.

Worked example · 200 W steady, smooth road, hoods position

Gradient	Rider + bike	Speed	Share of power lifting you	Height gained per hour
5%	85 kg	≈ 14.7 km/h	≈ 85%	≈ 734 m
8%	85 kg	≈ 10.0 km/h	≈ 92%	≈ 796 m
8%	75 kg	≈ 11.2 km/h	≈ 92%	≈ 894 m

8%, 85 kg: gravity = 85 × 9.81 × sin(arctan 0.08) ≈ 66 N. At 2.77 m/s that's ≈ 184 W of the 200 W. Height per hour = v sin θ × 3,600 ≈ 796 m.

Pacing a climb

Steady beats surges: because drag barely matters uphill, extra watts go almost entirely into climbing, but a surge you can't sustain costs you later.
Gear down early to keep a comfortable cadence (Ch 4).
Descents: the height you gain becomes speed on the way down. Brake before bends, not in them (Ch 5).

A 70 kg rider holds 210 W. What's their power-to-weight ratio?

210 ÷ 70 = 3.0 W/kg (using rider mass; the lab uses rider + bike).

↑ Back to the challenge

CH 4

Gears & cadence

Gears don't make power. They trade force for pedalling speed.

The gear ratio is chainring teeth ÷ cog teeth: how many times the rear wheel turns per pedal revolution. Multiply by the wheel's circumference and you get how far one pedal stroke moves you. Then speed = distance per stroke × cadence.

Worked example · distance per pedal turn and speed at 90 rpm

Bike and gear	Ratio	Per pedal turn	Speed at 90 rpm
Road, 50 × 11 (top gear)	4.55	≈ 9.5 m	≈ 51 km/h
Road, 34 × 28 (climbing gear)	1.21	≈ 2.6 m	≈ 14 km/h
MTB, 32 × 10	3.20	≈ 7.4 m	≈ 40 km/h
MTB, 32 × 50 (lowest)	0.64	≈ 1.5 m	≈ 8 km/h

Road wheel circumference taken as 2.1 m; 29-inch MTB wheel as 2.3 m. 50 ÷ 11 × 2.1 ≈ 9.55 m; × 90 per minute × 60 ÷ 1,000 ≈ 51.5 km/h. Real circumferences vary with tyre size.

Why cadence matters

For a given power, a higher cadence means less force on each pedal stroke (P = torque × pedalling speed). Spinning a lower gear is gentler on your knees and lets you respond to changes in gradient. Shift before the climb gets steep, while you can still ease the pressure on the pedals.

At 80 rpm in a gear that moves you 5 m per pedal turn, how fast are you going?

5 × 80 = 400 m per minute = 24 km/h.

↑ Back to the challenge

CH 5

Cornering & braking

Two thin contact patches handle every turn and every stop.

Stopping distance = reaction distance + braking distance: d = v t_react + v² ÷ (2a). Under braking, weight transfers to the front wheel, so the front brake does most of the work, until the rear lifts. A rough limit: maximum deceleration ≈ g × (horizontal distance from your centre of mass to the front contact patch ÷ its height).

Worked example · stopping at 0.5 g after a 1-second reaction

Speed	Reaction distance	Braking distance	Total
20 km/h	≈ 5.6 m	≈ 3.1 m	≈ 8.7 m
30 km/h	≈ 8.3 m	≈ 7.1 m	≈ 15.4 m
40 km/h	≈ 11.1 m	≈ 12.6 m	≈ 23.7 m

30 km/h = 8.33 m/s; braking = 8.33² ÷ (2 × 0.5 × 9.81) ≈ 7.1 m. Tip-over limit example: centre of mass 1.0 m high and 0.6 m behind the front contact patch gives about 0.6 g. On gravel or a wet road, grip can be far lower.

Cornering

To follow a bend of radius r at speed v you lean at tan φ = v² ÷ (g r): at 10 m/s on a 20 m bend, about 27°; at 8 m/s on a 10 m bend, about 33°. Braking and cornering share the same tyre grip, so brake in a straight line before the bend and release as you lean.

On the road

Wear a helmet, use front and rear lights after dark, ride with the traffic and be predictable. Check brakes and tyres before every ride.

Doubling your speed multiplies braking distance by how much?

d ∝ v², so by 4.

↑ Back to the challenge

CH 6

Trails, tyres & suspension

Off-road, the ground fights back. Tyres and suspension decide how much.

On dirt, tyres sink and knobs squirm, so rolling resistance rises steeply. Low pressures let the tyre wrap around rocks and roots for grip; suspension and bent legs spread out impacts so the wheels stay on the ground.

Worked example · surface vs speed at 200 W, 85 kg, upright position (from our Power Lab)

Surface	C_rr (illustrative)	Speed	Power to rolling
Smooth road	0.005	≈ 27.9 km/h	≈ 16%
Gravel	0.012	≈ 25.6 km/h	≈ 36%
Dirt trail	0.025	≈ 21.4 km/h	≈ 62%

Drops and landings

Worked example · average deceleration while landing a drop

Drop height	Landing speed	Stopped over	Average deceleration
0.5 m	≈ 3.1 m/s	0.25 m	≈ 2 g
1.0 m	≈ 4.4 m/s	0.25 m	≈ 4 g
1.0 m	≈ 4.4 m/s	0.40 m	≈ 2.5 g

Landing speed = √(2 g h); average deceleration = h ÷ stopping distance × g (vertical drop onto flat ground). Landing on a downslope (a proper landing ramp) turns much of that into forward speed instead.

Learn trail skills properly

Walk features before riding them, start small, wear a helmet (and gloves and pads for technical trails), and don't ride remote trails alone. A course such as NIM's mountain terrain biking course is a good place to start.

How much faster do you land from a 2 m drop than from a 0.5 m drop?

v ∝ √h, so √4 = 2 times faster (about 6.3 m/s against 3.1 m/s), with four times the energy.

↑ Back to the challenge

Every watt goes somewhere. Learn where.

How well do you know the physics of cycling?

Six chapters. Every number worked out.

Set your watts, the slope, the surface and your position, and see where every watt goes

Power & the three resistances

Air drag, position & drafting

Drafting

Climbing & watts per kilo

Pacing a climb

Gears & cadence

Why cadence matters

Cornering & braking

Cornering

On the road

Trails, tyres & suspension

Drops and landings

Learn trail skills properly

Regular rides → first brevet → multi-day & ultra → mountain biking → racing → coaching

Who sets the rules

Global

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India

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Audax India Randonneurs (AIR)

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