A car covers $$AB$$ distance with first one-third at velocity $$v_1$$ m s$$^{-1}$$, second one-third at $$v_2$$ m s$$^{-1}$$ and last one-third at $$v_3$$ m s$$^{-1}$$. If $$v_3 = 3v_1$$, $$v_2 = 2v_1$$ and $$v_1 = 11$$ m s$$^{-1}$$, then the average velocity of the car is ______ m s$$^{-1}$$.

Let the total distance $$AB$$ be $$3s$$.

Total time interval: $$t = t_1 + t_2 + t_3 = \frac{s}{v_1} + \frac{s}{v_2} + \frac{s}{v_3}$$

Given velocity relations: $$v_2 = 2v_1,\quad v_3 = 3v_1$$

Average velocity:

$$v_{\text{avg}} = \frac{3s}{t} = \frac{3s}{\frac{s}{v_1} + \frac{s}{2v_1} + \frac{s}{3v_1}}$$

$$\implies v_{\text{avg}} = \frac{3}{\frac{1}{v_1} \left(1 + \frac{1}{2} + \frac{1}{3}\right)} = \frac{3v_1}{\frac{6 + 3 + 2}{6}} = \frac{18v_1}{11}$$

$$v_{\text{avg}} = \frac{18 \times 11}{11} = 18\ \text{m/s}$$