Question

An object is falling freely under the gravitational force. Its velocity after traversing a distance h is v. If v depends upon gravitational acceleration g and distance. Prove with the help of dimensional analysis that v = kâ(gh), where k is a constant.

Answer 1

Answer:

v = k √gh

Explanation:

v depends upon gravitational acceleration g and distance

V = Velocity has Dimensional formula = Distance / Time = LT⁻¹

g = acceleration due to Gravity has dimensional formula = LT⁻²

Distance = h   has dimensional formula = L

as  velocity depending upon g & h only and

V has in dimensional formula  T⁻¹   and T is there only in g but T⁻²

so there must be Square root of  g so that both sides have T⁻¹

=> v ∝ √g  

LHS Dimensional formula = LT⁻¹

RHS became because of  √g   = L^(1/2)T⁻¹

so to match We need L^(1/2)  so h should have square root

=> v ∝ √gh  

=> v = k √gh   k is a constant

Answer 2

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