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Discussion
Every object in the solar system is pulled by the gravitational force F of the sun, which is given by
where k is a constant, m is the mass of the object, and d is the distance between the object and the sun.
At a given distance, how can we model the relationship between F and m.
Since d is fixed, the quantity d2k is constant. Writing
shows a linear relationship between F and m of the form
with slope d2k and intercept 0.
In other words, for any mass m, the force F is always m multiplied by the same constant d2k.
In this example, we cannot simply reduce distant to a constant because it is an important variable that will determine the force of gravity on an object. Instead, we can make the distinction that if m alone changes, F will change in direct proportion to m.
Direct Proportion
Directly proportional quantities are constant multiples of each other. In the context of modelling, we typically say, "y varies directly with xn," which means y=kxn for some constant k. This can be denoted y∝xn.
If y is directly proportional to xn, then x=0⟺y=0.
If y is directly proportional to xn, then if x increases (or decreases) by a factor of c, y increases (or decreases) by a factor of cn.
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It is given that y∝x5. Find the factor by which y increases when x increases by a factor of 2.
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Discussion
Using the same equation F=d2km, at a fixed mass m, how is the gravitational force F related to the distance d between the object and the sun?
Since m is fixed, write
so we see
Now check what happens when d is scaled by various factors n. Replace d by nd:
In particular:
Conclusion: multiplying the distance by n multiplies the force by 1/n2. Hence the gravitational force falls off as the square of the distance. Likewise, if the distance reduces by a fraction n, the gravitational force increases by the square of the inverse of n.
The relationship between F and d has some similarities between the relationship of F and m. Specifically, since there are multiple variables that cannot be fixed, both d and m are best described through their proportionality to F. However, F and m are directly proportional because as m increases, F increases. F and d clearly have a different relationship because as d increases, F decreases. Equivalently, we can say as d1 increases, F increases.
Inverse proportion
If y varies inversely with xn, then y=xnk.
If y is inversely proportional to xn (y∝xn1), then the y-axis is an asymptote of the graph of y=f(x).
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It is given that y∝x31. Find the factor by which y increases or decreases when x decreases by a factor of 51.
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