The velocity v of a particle at time t is giv | vedclass

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Question

(A) According to the principle of homogeneity of dimensions,the dimensions of each term in the equation must be the same.$1$. For the term $at$: $[at]

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$LT^{-2}, L$ and $T$

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(A) According to the principle of homogeneity of dimensions,the dimensions of each term in the equation must be the same.$1$. For the term $at$: $[at] = [v] = [LT^{-1}]$.Therefore,$[a] = \frac{[LT^{-1}]}{[T]} = [LT^{-2}]$.$2$. For the term $\frac{b}{t + c}$: Since $c$ is added to $t$,the dimension of $c$ must be the same as the dimension of $t$.Therefore,$[c] = [T]$.$3$. For the term $\frac{b}{t + c}$: The dimension of the whole term must be equal to the dimension of velocity $[v]$.$[\frac{b}{t + c}] = [v] \implies \frac{[b]}{[T]} = [LT^{-1}]$.Therefore,$[b] = [LT^{-1}] 	imes [T] = [L]$.Thus,the dimensions are $[a] = [LT^{-2}]$,$[b] = [L]$,and $[c] = [T]$.

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. What are the dimensions of $a, b,$ and $c$ respectively?

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