According to the principle of homogeneity of dimensions, only physical quantities with the same dimensions can be added or subtracted .

1. Dimension of $c$: In the equation, $c$ is added to $t$ (time) in the denominator of the second term. Therefore, the dimensions of $c$ must be the same as that of time $t$. So, $[c] = [T]$.
2. Dimension of $a$: Each additive term in the equation must have the same dimensions as the quantity on the left-hand side, which is velocity $v$ $[L T^{-1}]$. For the term $at$: $[a][t] = [v] \Rightarrow [a][T] = [L T^{-1}] \Rightarrow [a] = \frac{[L T^{-1}]}{[T]} = [L T^{-2}]$.
3. Dimension of $b$: For the term $\frac{b}{t+c}$, the entire term must have the dimensions of velocity. So, $\frac{[b]}{[t+c]} = [v]$. Since $[t+c] = [T]$, we have $\frac{[b]}{[T]} = [L T^{-1}] \Rightarrow [b] = [L T^{-1}] \times [T] = [L]$.

Thus, the dimensions of $a$, $b$, and $c$ are $[L T^{-2}], [L]$, and $[T]$ respectively.