The velocity v of a particle depends upon the time t according to the equation, v=a+bt+cd+t. Write the dimensions of a, b, c and d.

The velocity $v$ of a particle depends upon the time t according to the equation, $v = a + b t + \frac{c}{d + t}$ . Write the dimensions of $a, b, c a n d d$ .

A
$[a] = [v], [b] = [{LT}^{2}], [c] = [L], [d] = [T]$
B
$[a] = [v], [b] = [{LT}^{- 2}], [c] = [L], [d] = [T]$
C
$[a] = [v], [b] = [{LT}^{- 2}], [c] = [L], [d] = [L]$
D
$[a] = [v], [b] = [{LT}^{- 3}], [c] = [L], [d] = [T]$

Solution:

From principle of homogeneity,

$[a] = [v] o r [a] = [{LT}^{- 1}] [b t] = [v] o r [b] = \frac{[v]}{[t]} = \frac{[{LT}^{- 1}]}{[T]}$

or,

$[b] = [{LT}^{- 2}]$

Similarly,

$[d] = [t] = [T]$

Further,

$\frac{[c]}{[d + t]} = [v] o r [c] = [v] [d + t]$

or,

$[c] = [{LT}^{- 1}] [T] o r [c] = [L]$

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