Question

$v = at{\text{ }} + {\text{ }}\dfrac{b}{{t + c}} + {\text{ }}{v_0}$, is a dimensionally valid equation. Obtain the dimensional formula for $a,b$ and $c$ where $v$ is velocity, $t$ is time and ${v_0}$ is the initial velocity.

Answer 1

Hint: When we add two variables, they should have the same dimensional formula. When we equalize two variables, again they should have the same dimensional formula. Use this property to relate the dimensional formula of the unknown variables with the dimensional formula of known quantities given in the question.

Formula used:
Dimensional formula of velocity: ${\text{[distance][time}}{{\text{]}}^{{\text{ - 1}}}} = L{T^{ - 1}}$

Complete step by step solution:
The dimensional formula of two variables when they are being added or equalized should be the same. Since the expression in the question is given to be dimensionally correct, we can use the previously mentioned principle.
On the right-hand side of the equation since we are adding $at$ to ${v_0}$ they must have the same dimensional formula. So the dimensional formula of a can be calculated as:
$\Rightarrow L{T^{ - 1}} = [a]T$
Dividing both sides by the unit of time, we get
$\Rightarrow [a] = L{T^{ - 2}}$ which is the dimensional formula of $a$
Similarly, in the second term on the right-hand side, since we are adding $c$ with the variable of time, it must have the units of time too. Hence
$\Rightarrow [c] = T$
The combined fraction $\dfrac{b}{{t + c}}$ must have the units of velocity since it is added to ${v_0}$ and since the denominator has the units of time, we can write
$\Rightarrow L{T^{ - 1}} = \dfrac{{[b]}}{T}$
$\Rightarrow [b] = L$ which is the dimensional formula of $b$.

Note:
To solve such problems, we must know the dimensional formulae of basic physical quantities such as the velocity of an object. We can only add and equal variables of the same quantity i.e. the same dimensional formula however quantities with different dimensional formula can be multiplied or divided by other and we cannot equalize the dimensional formula of these variables.