The speed `v` of a satellite moving in a circular orbit around the earth depends on the gravitational constant `G`, mass of the earth `m_(e)` and radius of circular orbit `r`. Estabish the relation using dimensions.

Explanation

`v = fn (G, m_(e), r)`
`v prop G^(a) M_(c )^(b) r^(c )`
`[LT^(-1)] prop [M^(-1) L^(3) T^(-2)]^(a) [M]^(b) [L]^(c )`
`M^(0) L^(1) T^(-1) prop M^(-a + b) L^(3a + c) T^(-2a)`
Comparing powers of `M, L` and `T`
`-a + b = 0`
`3a + c = 1`
`- 2a = - 1`
`a = (1)/(2), b = - (1)/(2), c = - (1)/(2)`
`v prop G^((1)/(2)) M_(e)^((1)/(2)) r^(-(1)/(2)) implies v prop sqrt((G M_(e))/(r)) implies v = k sqrt((G M_(e))/(r))`
From experiment, `k = 1`
`v = sqrt((G M_(e))/(r))`