Question

8. Prove with the help of dimensional analysis that the
st
equation h=1/2 gt for the distance travelled by a
body falling freely under gravity in time t is incorrect.
Find the correct equation with the help of dimensions.​

Answer 1

Answer:

For dimensionally correct formula, dimensions at LHS = Dimensions at RHS

Here, h = 1/2gt

[h] = [g][t]

[M⁰L¹T⁰] = [M⁰L¹T⁻²] [M⁰L⁰T¹]

[M⁰L¹T⁰] ≠ [M⁰L¹T⁻¹]

So, it's dimensionally incorrect.

[h] =k [g]ᵃ[t]ᵇ where k is unit less constant

[M⁰L¹T⁰] =  [M⁰L¹T⁻²]ᵃ [M⁰L⁰T¹]ᵇ

[M⁰L¹T⁰] =  [M⁰LᵃT⁻²ᵃ⁺ᵇ]

Equating LHS and RHS,

a = 1

a-2a+b = 0

or,-2 + b = 0

or, b= 2

So, required equation is

h = kgt²

or,  h = 1/2gt² (We can't get the numeric value of the equation through dimensional analysis, I have added according to the question but the correct answer is h = kgt²)