Question

the velocity of freely falling body is a function of the distance fallen through (h) and acceleration due to gravity (g). show by method of dimensions that v=k under root gh​

Answer 1

we have to show $v=k\sqrt{gh}$ by the method of dimensions.

where v is velocity of freely falling body

k is constant, g is acceleration due to gravity and h is height.

dimension of v = $LT^{-1}$

dimension of g = $LT^{-2}$

dimension of h = $L$

from dimension rule,

we can write it, $v=kg^xh^y$

or, $[LT^{-1}]=k[LT^{-2}]^x[L]^y$

or, $[LT^{-1}]=k[L]^{(x+y)}][T]^{(-2x)}$

comparing both sides,

x + y = 1, .....(1)

-2x = -1 => x = 1/2

from equation (1), y = 1/2

hence, $v=kg^{1/2}h^{1/2}=k\sqrt{gh}$ hence proved.

Answer 2

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