According to the principle of homogeneity of dimensions, only quantities with the same dimensions can be added, subtracted, or equated. From the given equation $v = at + \frac{b}{t+c}$:

1. In the denominator, $c$ is added to time $t$. Therefore, the dimension of $c$ must be the same as that of time $t$. So, $[c] = [T]$.
2. The dimension of the term $at$ must be equal to the dimension of velocity $v$. So, $[a][T] = [L T^{-1}] \implies [a] = [L T^{-2}]$.
3. The dimension of the term $\frac{b}{t+c}$ must also be equal to the dimension of $v$. So, $\frac{[b]}{[T]} = [L T^{-1}] \implies [b] = [L]$. Therefore, the dimensions of $a$, $b$, and $c$ are $[L T^{-2}]$, $[L]$, and $[T]$ respectively .