Question

The position of a particle at time t is given by the relation $x(t) = \left( \frac{v_{0}}{\alpha} \right)(1 - c^{- \alpha t}),$ where $v_{0}$ is a constant and $\alpha > 0$. The dimensions of $v_{0}$ and $\alpha$ are respectively

• A. $M^{0}L^{1}T^{- 1}$and$T^{- 1}$
• B. $M^{0}L^{1}T^{0}$ and $T^{- 1}$
• C. $M^{0}L^{1}T^{- 1}$and$LT^{- 2}$
• D. $M^{0}L^{1}T^{- 1}$ and $T$

Answer 1

Answer: (A)

$M^{0}L^{1}T^{- 1}$and$T^{- 1}$

Answer 2

From the principle of dimensional homogeneity $\lbrack\alpha t\rbrack$

= dimensionless $\therefore\lbrack\alpha\rbrack = \left\lbrack \frac{1}{t} \right\rbrack = \lbrack T^{- 1}\rbrack$

Similarly $\lbrack x\rbrack = \frac{\lbrack v_{0}\rbrack}{\lbrack\alpha\rbrack}$ $\therefore\lbrack v_{0}\rbrack = \lbrack x\rbrack\lbrack\alpha\rbrack = \lbrack L\rbrack\lbrack T^{- 1}\rbrack = \lbrack LT^{- 1}\rbrack$.