Suppose we are told that the acceleration of a particle moving in a circle of radius r withuniform speed v is proportional to some power of r, say r^n , and some power of v, say v^m , determine the powers of r and v

I would look at the units.

r . . . meters
v . . . meters/second
and
acceleration . . . meters/second^2 .

1). The only way to get sec^2 on the bottom is to use v^2.
That gives you (meters^2)/(sec^2).

2). If you divide that by meters,
you have (meters)/(sec^2), and THERE'S your acceleration !

So we know that (v^2)/(r) is acceleration.

In terms of the way the question is written:

v^m = v^2. m = 2 .

r^n = 1/r. r = -1 .

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