Math Help - combination of functions

1. combination of functions

Hello, I have this one question that I have been mulling over for a day now and I cant seem to figure it out. I have to develop a formula that predicts the velocity of a cyclist and in the question it states that the speed depends on three independant factors, these are; the speed at which the cyclist peddles to turn the front gear, in rpm, the gear ratio from the front gear to the rear gear, a ratio between the number of teeth on the front, compared to number of teeth in the rear, and final, the diametre of the wheel. So far, I've tried to figure this out and im stuck, anyone help me.

2. Developing a formula

Hello walk_in_fark
Originally Posted by walk_in_fark
Hello, I have this one question that I have been mulling over for a day now and I cant seem to figure it out. I have to develop a formula that predicts the velocity of a cyclist and in the question it states that the speed depends on three independant factors, these are; the speed at which the cyclist peddles to turn the front gear, in rpm, the gear ratio from the front gear to the rear gear, a ratio between the number of teeth on the front, compared to number of teeth in the rear, and final, the diametre of the wheel. So far, I've tried to figure this out and im stuck, anyone help me.
Let's use some letters to stand for the three variables that you describe:

• The cyclist pedals at $P$ rpm.

• The ratio of the number of teeth on the front gear to the rear gear is $n:1$.

• The diameter of the rear wheel is $d$ metres,

So in one minute, the pedals, and therefore the front gear, complete $P$ revolutions. Since the gear ratio is $n:1$, the rear gear will need to complete $n$ revolutions for every one revolution that the front gear makes. This is because the number of gear teeth that move by a fixed point must be the same at the front and rear. So the rear gear - and hence the rear wheel - makes $nP$ revolutions in one minute.

Now if the diameter of the rear wheel is $d$ metres, its circumference is $\pi d$ metres, which is, of course, the distance the cycle will move forward in one revolution of the rear wheel. So the total distance that it moves in one minute is $nP\times\pi d$ metres. If we divide by 60, that will give us a speed of

$\frac{\pi dnP}{60}$ metres per second.