Basic question about basic concept: proportionality

[Ontophile]

This is not homework. I'm reading a physics for laypeople book and I'm trying to wrap my head around something that other people obviously have little or no trouble understanding.

I understand what it means when someone says that a body's acceleration is directly proportional to the magnitude of the impressed force. It means that if you apply one more unit of force, you'll get one more unit of acceleration out of it. Double the force applied, double the resulting acceleration; triple the force, triple the acceleration.

What I don't understand is what to imagine when someone says that something is directly proportional to the square of something, or directly proportional to the square root of something. Is being "directly proportional to the square root" of something the same as being "inversely proportional to the square" of something?

1) Suppose we lived in a universe in which a body's acceleration was directly proportional to the square of the impressed force. What would that mean? Would it mean that if I doubled the impressed force, I'd quadruple the resulting acceleration? Or would it mean that if I wanted to double the acceleration, I'd have to quadruple the impressed force?

2) Likewise, say a body's acceleration was directly proportional to the square root of the impressed force. What would that mean?

3) Suppose that tripling the impressed force yielded 6 times the acceleration, and that quadrupling the force yielded 8 times the acceleration. What sort of proportional relationship would that be?

4) Suppose that adding 1 unit of force yielded 1 unit of acceleration, and that adding 2 units of force yielded 1.5 units of acceleration, and that adding 3 units of force yielded 1.8333... units of acceleration, and that adding 4 units of force yielded 2.08333... units of accceleration. What sort of proportional relationship would that be?

5) Suppose that adding 1 unit of force yielded 1 unit of acceleration, and that adding 2 units of force yielded 1.5 units of acceleration, but that adding 3 units of force yielded 1.75 units of acceleration, and that adding 4 units of force yielded 1.875 units of acceleration. What sort of proportional relationship would that be?

Thanks in advance for your help...

 

[sophiecentaur]

'Directly proportional' is a verbal description for the algebraic equation:
y = ax,
in which y is said to be directly proportional to x.

Otoh, if you see the expression
y=ax+b,
you would say that y is proportional to x - i.e it gets bigger as x gets bigger but will not equal zero when x is zero. 'Directly' means there is no extra 'constant' term (b).

If you say that y is directly proportional to x², that would be written as
y=ax²

or, if y is directly proportional to root x, then

y=a√x

Using the maths may bring on the pains but is nicely precise and you can tap it into your calculator if you know the numbers.

If you want to find the equation for some process by looking at some measured results, that is a bit harder because you, effectively, have to try to 'solve' an equation for which there is not always a simple solution or a simple expression to show the relationship. Scientists are always being presented with this sort ofd problem and sometimes there are horrific sums needed to produce a formula.