Question

An object is falling freely under the
gravitational force. Its velocity after
travelling a distance h is v. If v depends
upon gravitational acceleration g and
distance, prove with dimensional
analysis that v=kgh where k is a
constant.

PLEASE ANSWER ASAP!!​

Answer 1

Given that,

An object is falling freely under the

gravitational force. Its velocity after

travelling a distance h is v. v depends

upon gravitational acceleration g and

distance h.

By principle of homogeneity, The dimensions on the both sides of a equation must be equal.

Given, v depends on g & h

We have, v ∝ gh [ As stated in the question]

Let say v = kg^a h^b [ To remove the proportionality, I have introduced a proportionality constant]

Dimensions of v (velocity) are LT⁻¹

Dimensions of Acceleration due to gravity are LT⁻²

Dimensions of Height h are L

Now we have,

LT⁻¹ = (LT⁻²)ᵃ * (L)ᵇ

LT⁻¹ = Lᵃ⁺ᵇ * T⁻²ᵃ

Equating both sides,

a + b = 1

-2a = - 1

a = 1/2

b = 1/2

Therefore, From dimensional analysis it can be written as

V = k √gh

Hence proved.