Question

v = `at + b/(t + c) + "v"_0` is a dimensionally valid equation. Obtain the dimensional formula for a, b and c where v is velocity, t is time and v₀ is initial velocity.

Answer 1

Given: v = `at + b/(t + c) + "v"_0`

As only dimensionally identical quantities can be added together or subtracted from each other, each term on R.H.S. has dimensions of L.H.S. i.e., dimensions of velocity.

∴ [L.H.S.] = [v] = [L¹T⁻¹]

This means, [at] = [v] = [L¹T⁻¹]

Given, t = time has dimension [T¹]

∴ [a] = `(["L"^1"T"^-1])/(["t"]) = (["L"^1"T"^-1])/(["T"^1]) = ["L"^1"T"^-2] = ["L"^1"M"^0"T"^-2]`

Similarly,

[c] = [t] = [T¹] = [L⁰M⁰T¹]

∴ `(["b"])/(["T"^1]) = ["v"] = ["L"^1"T"^-1]`

∴ [b] = [L¹T^-1] × [T¹] = [L¹] = [L¹M⁰T⁰]