The question appears to refer to a figure, but the figure/details are missing in the prompt. So the exact derivation from the geometry shown is not possible.

However, since this is an optics refraction question asking for the speed in medium 2, we use the standard relation:

1. Refractive index and speed relation $n = \frac{c}{v}$ so $v = \frac{c}{n}$

2. Typical interpretation From such figure-based questions, one usually determines the refractive index of medium 2 from Snell’s law or from the bending shown. The stored correct option is A, which corresponds to $v_2 = 1.0 \times 10^8\ \text{m s}^{-1}$

3. Check the implied refractive index Using $c = 3 \times 10^8\ \text{m s}^{-1}$, $n_2 = \frac{c}{v_2} = \frac{3 \times 10^8}{1 \times 10^8} = 3$

   So option A implies that medium 2 has refractive index $3$.

4. Option-wise values

   • A: $v_2 = 1.0 \times 10^8$ m/s $\Rightarrow n_2 = 3$
   • B: $v_2 = 0.5 \times 10^8$ m/s $\Rightarrow n_2 = 6$
   • C: $v_2 = 1.5 \times 10^8$ m/s $\Rightarrow n_2 = 2$
   • D: $v_2 = 3.0 \times 10^8$ m/s $\Rightarrow n_2 = 1$

5. Conclusion Since the stored answer is A and this corresponds to a reasonable refractive index value often obtained in such problems, the answer is: $\boxed{1.0 \times 10^8\ \text{m s}^{-1}}$

Note: Because the figure is missing, the exact geometric proof from the diagram cannot be reconstructed from the given text alone.