Is there a formula to determine how gear ratios effect speed?

[Tornado]

I was hacking Sports Car GT, and I figured that if there was a mathematical way to determine what the top speed was based purely on the gear ratios than it would be easier than doing it how I've been, which is basically blind guessing backed up by begrudging testing. I've also seen various games use what appears to be a mathematically based way of determining speed within tuning menus, so I assume there is in fact such formula.
I'm basically asking if there is some kind of formula that determines what speed a car would be able to get to in each gear based on gear ratios alone, or (if that's not possible) gear ratios combined with tire sizes.

 

[Venari]

Rather than a formula, I'm a sucker for working it out.

Take your tyre size, say it's a 19" rim with a 275/35 tyre. Well, that's a rolling circumference of 2045mm, there or thereabouts. (Trust me.)

So, for every one revolution at the rim, you travel 2045mm, 2.045m, or 0.00245 km.

For every one rpm that wheel spins, you travel at 2.045m/min or 0.002045km/min. If you travel that much in a min, then 60x that much in an hour... = 0.1227 km/hour.

Now, spin that up to 1000rpm, and you're travelling at 122.7 km/hour = or around 75mph.

Got a final drive ratio? Well, plug that in. Say 3.540:1. So your input shaft to the diff is spinning at 3540rpm, and gets reduced to 1000rpm.

If you have a gear ratio in your gearbox of 1:1, then in that gear, your engine is doing 3540 rpm, the output shaft from the box is doing 3540 rpm, your axleshaft is doing 1000rpm, and the car is travelling at 122.7km/hour.

If you've got a gear ration of 2.000:1, then you're spinning the engine at 7080rpm, the output from the gearbox is 3540rpm... etc.



Now.

Say the car can rev to 8,000rpm in top, and top ratio is 0.900:1

So, juggle the numbers a bit. 8,000rpm input, 8,000/0.9 = 8,888rpm output speed.

Now divide that by your final drive ratio, 3.54 to get 2511rpm at the axles. 2511rpm/1000rpm we used for our example = 2.511 * 122.7kmh = 308kmh, or about 190mph.


The key is knowing your tyre's rolling circumference. www.toyo.co.uk used to have lots of tyre details including rolling radii. From GT4, you'll have to guess an approximate tyre size from the vehicle - or look it up on the 'net. My FTO has 205/50R16 tyres, for example, and I know the GT4 ratios and speeds are (fairly) accurate.

 

[Boundary Layer]

Gah, he beat me.... Which is fine because it saves me the trouble of typing it out in too much detail.

Basically though, the formula is as follows:
Driven gear A.V = ((Driver gear diameter)/(Driven gear diameter))*Driver gear A.V

or
​

Driven gear A.V = (Gear Ratio^-1)*Driver gear A.V

where A.V represents angular velocity
​

And as Venari pointed out already, determing the velocity is just a matter of multiplying your tire diameter by its angular velocity.

...or you could save yourself a hassle and go here:
http://www.grimmjeeper.com/gears.html