y = axn and y = a/xn, the functions of direct and indirect proportionality of higher order
For corresponding pairs of values, we have drawn the conclusion from
| y=mx: | that | y1/x1=y2/x2=y3/x3=···=m | y=m/x¹: | that | y1x1=y2x2=y3x3=···=m; |
similarly, we have now quite generally
| y=axn: | y1/x1n=y2/x2n=y3/x3n=···= a | y=a/xn: | y1x1n=y2x2n=y3x3n=···=a |
| y1/x1n = y2/x2n | yields the proportion | y1/ y2 =x1n/x2n | ||
| y1x2n = y2x2n | yields the proportion | y1/y2 = x2n/x1n |
As before, we speak of direct proportionality and indirect proportionality, that is, of higher order proportionality.
Conversely, if you have discovered such a proportionality, then the preceding statements show that the functions must have the form y = axn or y = a/xn
y = axn is the function of direct, higher order proportionality.
y = a/xn is the function of indirect, higher order proportionality.
Example 1. A sphere running down an inclined tube covered during the last seconds (t) the distances (s) in cm:
| t = x | 1 | 2 | 3 | 4 | 5 | ··· | ||||||
| s = y | 5 | 20 | 45 | 80 | 125 | ··· |
We examine whether y/x² is constant and discover that
5/1² = 20/2² = 45/3² = 80/4² = 125/5² = 5.
The distance is directly proportional to the square of the time. Such statements are made in Physics. In general, we conclude now that there exists a second order proportionality between the path (s = y) and the time (t = x). The function must have the form y = ax² or s = at². A single corresponding pair of values finds that a = 5. The required equation is s = 5t².
Example 2. Let the force of attraction k between two masses m1 and m2 be:
1) proportional to
the mass m1,
2) proportional to the mass m2,
3) inversely proportional to the square of the
distance r between the masses.
Find the function describing this phenomenon.
| k=cm1 | c=c1m2 | c1=c2/r² | whence | k = cm1=c1m2cm1=c2cm1m2/r² |
Newton's gravitational law of attraction is: k = c1c2m1m2/r² .
The determination of the constant c by a pair of values is not easy, but he succeeded. If the Earth is the one mass, you speak of Earth's gravitational force. The same law holds between electric and magnetic charges, but the constant c has then another value.