A person's cycling rate can be expressed mathematically in the equation that describes their cycling motion. Specifically, the cycling rate can be represented by the frequency or number of revolutions per unit time (usually in seconds) that the person completes while cycling.
The equation that describes a person's cycling motion is typically a kinematic equation that relates the person's position, velocity, and acceleration as a function of time. One commonly used equation is:
d = vit + 1/2at^2
where d is the distance traveled by the person, vi is the initial velocity (usually zero), a is the acceleration, and t is the time elapsed.
The cycling rate can be incorporated into this equation by expressing the velocity as a function of the frequency of revolutions (f) and the radius of the wheel (r). This gives:
v = 2πrf
where v is the velocity of the cyclist, r is the radius of the wheel, and π is the mathematical constant pi (approximately 3.14).
Substituting this expression for v into the kinematic equation gives:
d = (2πrf)t + 1/2at^2
This equation shows how the person's cycling rate, represented by the frequency f, affects their distance traveled as a function of time. If the person increases their cycling rate, their velocity increases, and they will travel a greater distance in the same amount of time.