The escape velocity, v, of a body depends on the acceleration due to gravity, g, on the planet and the radius of the planet, R. Using the method of dimensions, we can obtain the relation between v, g, and R.
From the given information:
We know that the dimensions of escape velocity are [LT^-1] (length/time).
The acceleration due to gravity, g, has dimensions [LT^-2] (length per time squared).
The radius of the planet, R, has dimensions [L] (length).
We know that the escape velocity depends on the square root of the product of the acceleration due to gravity and the radius of the planet.
Therefore, we can express the relation using dimensions as follows:
v ∝ √(gR)
We can add a constant to the relation to make it dimensionally correct, so we can write:
v = k √(gR)
Where k is a dimensionless constant.
So, using the method of dimensions, the relation between v, g, and R can be given as v = k √(gR).
