Electrons in a certain energy level \(n=n_{1}\), can emit 3 spectral lines. When they are in another energy level, \(n=n_{2}\) they can emit 6 spectral lines. The orbital speed of the electrons in the orbits are in the ratio

Number of emitted spectral lines
\[
\begin{aligned}
N &=\frac{n(n-1)}{2} \\
& \text { 1st case } & &=3 \\
& \therefore & & \\
\Rightarrow & &=\frac{n_{1}\left(n_{1}-1\right)}{2} \\
\Rightarrow \quad\left(n_{1}-3\right)\left(n_{1}+2\right) &=0 \\
n_{1}=3, n_{1} &=-2
\end{aligned}
\]
Negative value of \(n_{1}\) is not possible
\[
\begin{aligned}
& & n_{1} &=3 \\
\text { 2nd case } & N &=6 \\
& \text { Again, } & 6 &=\frac{n_{2}\left(n_{2}-1\right)}{2} \\
\Rightarrow \quad & n_{2}^{2}-n_{2}-12 &=0 \\
\Rightarrow \quad\left(n_{2}-4\right)\left(n_{2}+3\right) &=0 \\
n_{2}=4, n_{2} &=-3
\end{aligned}
\]
Again, as \(n_{2}\) is always positive
\(\therefore \quad n_{2}=4\)
Velocity of electron \(v=\frac{Z e^{2}}{2 \varepsilon_{0} h n}\)
\(\therefore \quad \frac{v_{1}}{v_{2}}=\frac{n_{2}}{n_{1}}\)
\(\Rightarrow \quad \frac{v_{1}}{v_{2}}=\frac{4}{3}\)