## [Bike] Gearing On The Back Of An Envelope

From: andrew cooke <andrew@...>

Date: Mon, 4 Oct 2021 21:28:20 -0300

Uphill

If you + bike weigh 100kg and you can put out a steady 100W then you can
ascend vertically at 0.1m/s (work against gravity if g=10m/s^2).

On a 10% incline (steep, but possible where I ride in the Andes foothills)
that means you can move forwards at 1m/s.

The circumference of a 700C wheel is about 2m and the slowest comfortable
cadence while seated is around 60rpm.  So with 1:1 gearing that's 2m/s.

So you need 1:2 gearing for the climb.

That kind of gearing is pushing the low end of available MTB gears.  In other
words, pretty much no-one has gearing that low.  Pretty much all entry-level
riders would have real problems climbing a 10% grade for any distance.

Downhill

Air resistance increases as v^3.  That has a sharp "bend" - either it's not
important or it's a brick wall.  So basically everyone (whatever their power
output) is limited to a similar top speed.  From experience it's roughly
60km/h.

Faster / steeper than that you need to get seriously aero and pedalling
efficiently isn't really an option.

You can probably manage 100rpm if you have to, in your top gear, all out.
With a 2m circumference wheel that's 200m/min.

60km/h is 1km/min.  So you need 5:1 gearing for the descent (to go from
200m/min to 1,000 m/min).

This is consistent with the top end of traditional "race" gearing (maybe 53
teeth at the front and 11 at the rear).

Gear Range

The above implies a gear range from 1:2 to 5:1.  That's a factor of 10, or
1,000%.

In practice, pro riders can put out around 4x the power / weight used above.
Compared to inexperienced riders they are absolute monsters.  That shifts the
lower gears needed up by a factor of 4.  So instead of a 1,000% range they
need a 250% range, which is well within typical "race bike" gearing.

Non-pro riders (and people with loaded touring bikes) that have the technical
understanding compromise at both ends.  At the top, 4:1 is generally enough,
and at the bottom they can live with maybe 1:1.5.  That gives a 600% range
which is possile with a Rohloff IG hub, or a frankenstien mix of road and MTB
components (as on my own road bike).

Currently fashionable 1x road / gravel can reach maybe 500% range.  Exactly
where that is "centred" depends on the front ring - there will likely be
significant compromise at both the high and low ends.

Andrew