The arcs traced by projectiles are approximated by parabolas. Parabolas come in all shapes and sizes - tall and thin, short and wide - or so I thought.
While looking at the graph of a parabola on Geogebra, I noticed that you could "flatten it out" by changing the x-scale on the axes or by zooming in (changing x- and y-scales proportionally) on the vertex. This is intuitive: if you zoom in on any smooth curve, it straightens out - in calculus you call that local linearity.
But this means that any two parabolas can be made to look identical just by zooming (as opposed to scaling only one axis), which is how two similar objects can be made to look identical.
We say two object are similar if their corresponding lengths form proportions (pairs of equal fractions); can we prove that two parabolas are similar in this way?
Say you have two parabolas, y = ax² and y = bx². To show that two shapes are similar, we find a scale factor (k) so that any length on the first shape equals k times the corresponding length on the second shape. For parabolas, let's use the pair of lengths x and y for convenience.
On the first parabola, this pair of lengths is x1 and ax1².On the second parabola, the pair of lengths is x2 and bx2².
We must find a k so that x2 = kx1 and bx2² = kax1².
Substituting for x2, we have b(kx1)² = kax1².
So bk²x1² = kax1².
Dividing, we have bk = a, or k = a/b.
So the scale factor between any two parabolas y = ax² and y = bx² is a/b, conveniently.