# Gear shifts and drive speed on bikes

If I am driving in the 5th gear, can I know what is my velocity?

Let's assume I am on a perfectly flat surface and moving monotonously.

• Can you count how many rotations of the cranks you do in a minute? Jan 13 '16 at 21:25
• Simplest approach is to simply buy an inexpensive bike speedometer. Jan 14 '16 at 3:32

For this you will need to know the gearing ratio from front to back (or number of teeth on each gear), the wheel diameter, and the rate at which you are pedaling.

1) RPM to rad/s: Firstly determine how fast the front gear is turning, assuming that you are pedaling at one complete revolution per 2 seconds: 30 RPM / 60 (convert from minutes to seconds) * 2pi rad/rev:

``````pedalSpeedRPM = 30
speedRadPerSec = pedalSpeedRPM * 2 * pi / 60
``````

2) Next determine how fast the rear wheel is spinning by using the gear ratio; the ratio of the number of teeth on the front ring to the number of teeth on the back ring. For example, if the front ring has 53 teeth and the rear ring has 13 teeth, then the ratio is 4.08. That means that the rear wheel is spinning 5.2 times faster than the pedals. Thus:

``````gearRatio = 4.08
``````

3) Rotational to linear motion: If you know the diameter of your wheel, you can determine how far it moves with one revolution, for example, a 26" wheel (in metric because that's what I know) has a radius of approximately 0.33m. Using the formula `velocity = radius x angular velocity` gives

``````wheelRadius = 0.33
velocityKPH = velocity * 3.6
``````

So to bring it all together, you need to know the following variables:

``````pedalSpeedRPM = 30
gearRatio = 4.08

Calculations:
speedRadPerSec = pedalSpeedRPM* 2 * pi / 60
velocityKPH = velocity * 3.6
``````

Running this on MATLAB gives the result:

``````velocityKPH =

15.2274
``````

Which isn't a bad estimate for middle gear, non-frantic pedaling.

Another example, with (maybe) more realistic numbers:

``````pedalSpeedRPM = 60; % Faster pedalling
gearRatio = 4.2; % Abitrary

velocityKPH =

35.1507
``````

So I guess to (finally) answer your question, yes, you can determine it, but you need to know the gearing ratio, your wheel size, and an estimate of your pedaling rate.

• Wouldn't 53x13 yield a gear ratio of 4.077? 53/13 = 4.077. If i use that gear ratio with your numbers it works out to 15.2 km/h, which is the same result I get with my formula. Jan 13 '16 at 21:40
• Woops, meant to be 19. Miscopied from the example. Thanks for that!
– Lui
Jan 13 '16 at 21:41
• 53/19 is an even smaller ratio at 2.79. Unless I really don't understand gear ratios. Jan 13 '16 at 21:43
• I was copying blindly. Should be fixed now. The wikipedia article I quickly found must be stating a different ratio.
– Lui
Jan 13 '16 at 21:44
• Pedal RPM is commonly called cadence and 30 is low Jan 13 '16 at 21:45

A simple way is to count wheel revolutions directly, this is how most cycle computers work, as you can ignore the gear you are in. If you don't have a cycle computer, paint a white mark on the front tire and count how many revolutions the tire does in a minute. Multiply the circumference of the tire by 60 to get speed. You could use the tire radius and calculate the circumference, or using the mark or tire valve, roll the tire along the ground and measure one full revolution (ideally with you weight on the bike).

An alternate (and more accurate) way to measure speed is from GPS - most cell phones have one.

If you don't feel like doing all these mental calisthenics on the bike, get a cheap speedo that runs off a magnet. Mount the readout on your handlebars and simply look at it to learn your speed.

No need to overcomplicate this.

You can calculate your speed on a bicycle based just on the gearing, and it doesn't depend on the slope of the road, but you are going to need more information than "5th gear".

You need to know a few things.

W. How many teeth on the front chainring of whatever gear you are on.

X. How many teeth on the rear cog of whatever gear you are on.

Y. The circumference of the rear tire, in CM

Z. The cadence at which you are pedalling, usually in RPM

Now that you have the required information, you plug it into the following formula.

W/X * (Y /100000) * Z * 60

Let's look at an example. Assuming you were in a gear with a 50 tooth chainring and a 16 tooth sprocket, pedalling at 90 RPM, and your bike rear wheel circumference of 210 cm (rough circumference of road bike tire) and are pedalling at 90 RPM, then your speed would be as follows.

50/16 * (210/100000) * 90 * 60 = 35 km/h

The 100,000 constant in the equation converts the diameter of the wheel to km, and the 60 constant in the equation converts RPM to revolutions per hour allowing us to calculate the speed in km/h. Hopefully somebody else can check this equation as I just derived it in my head, but the numbers seem accurate based on my experience with riding.

• Pedantic correction story: I have ridden at about 50km/h at about minus 90rpm of the cranks. Jan 13 '16 at 21:29
• @Vorac we're measuring speed not velocity, so only magnitude matters.
– Móż
Jan 13 '16 at 21:36
• @Mσᶎ, lol, true. I was referring to the freehubs and gravitational acceleration. Jan 13 '16 at 22:08
• @vorac - the question specified "perfectly flat surface"
– Criggie
Jan 14 '16 at 1:54