Question
The velocity $$v$$ of a particle at time $$t$$ is given by $$v = at + \frac{b}{{t + c}},$$ where $$a,b$$ and $$c$$ are constants. The dimensions of $$a,b$$ and $$c$$ are respectively
A.
$$\left[ {L{T^{ - 2}}} \right],\left[ L \right]\,{\text{and}}\,\left[ T \right]$$
B.
$$\left[ {{L^2}} \right],\left[ T \right]\,{\text{and}}\,\left[ {L{T^2}} \right]$$
C.
$$\left[ {L{T^2}} \right],\left[ {LT} \right]\,{\text{and}}\,\left[ L \right]$$
D.
$$\left[ L \right],\left[ {LT} \right]\,{\text{and}}\,\left[ {{T^2}} \right]$$
Answer :
$$\left[ {L{T^{ - 2}}} \right],\left[ L \right]\,{\text{and}}\,\left[ T \right]$$
Solution :
$$\eqalign{
& {\text{The given expression is }}v = at + \frac{b}{{t + c}} \cr
& {\text{From principle of homogeneity}} \cr
& \left[ a \right]\left[ t \right] = \left[ v \right] \cr
& \left[ a \right] = \frac{{\left[ v \right]}}{{\left[ t \right]}} = \frac{{\left[ {L{T^{ - 1}}} \right]}}{{\left[ T \right]}} = \left[ {L{T^{ - 2}}} \right] \cr
& {\text{Similarly, }}\left[ c \right] = \left[ t \right] = \left[ T \right] \cr
& {\text{Further,}}\,\frac{{\left[ b \right]}}{{\left[ {t + c} \right]}} = \left[ v \right] \cr
& {\text{or,}}\,\left[ b \right] = \left[ v \right]\left[ {t + c} \right] \cr
& {\text{or,}}\,\left[ b \right] = \left[ {L{T^{ - 1}}} \right]\left[ T \right] = \left[ L \right] \cr} $$
NOTE
If a physical quantity depends on more than three factors, then relation among them cannot be established, because we can have only three equations by equalising the powers of $$M, L$$ and $$T.$$