1. Use Bohr model speed formula

   For a hydrogen-like atom, the speed of the electron in the $n$th orbit is $v_n = \frac{Z\alpha c}{n}$ where:

   • $Z$ = atomic number
   • $n$ = orbit number
   • $\alpha$ = fine structure constant

   Hence, $v_n \propto \frac{Z}{n}$

2. For $\mathrm{He}^+$ in the 3rd orbit

   Helium ion $\mathrm{He}^+$ is hydrogen-like with $Z=2, \quad n=3$ So, $v_{\mathrm{He}^+} \propto \frac{2}{3}$

3. For hydrogen atom in the 3rd orbit

   For hydrogen, $Z=1, \quad n=3$ So, $v_{\mathrm{H}} \propto \frac{1}{3}$

4. Take the ratio

   $\frac{v_{\mathrm{He}^+}}{v_{\mathrm{H}}} = \frac{2/3}{1/3} = 2$

   Therefore, the required ratio is $2:1$

5. Option check

   • A: $1:1$ ❌
   • B: $1:2$ ❌
   • C: $4:1$ ❌
   • D: $2:1$ ✅

Therefore, the correct answer is D.