1. Kinetic energy in Bohr model

For a hydrogen-like atom, the total energy of the electron in the $n^{\text{th}}$ orbit is $E_n = -\frac{13.6 Z^2}{n^2}\,\text{eV}$ where $Z$ is the atomic number.

In the Bohr model, $K_n = -E_n = \frac{13.6 Z^2}{n^2}\,\text{eV}$ so the kinetic energy of the electron in orbit $n$ is directly proportional to $\dfrac{Z^2}{n^2}$.

2. Find kinetic energy for H atom at $n=2$

For hydrogen, $Z=1$. Thus, $K_{H,n=2} = \frac{13.6(1)^2}{2^2} = \frac{13.6}{4}$

3. Find kinetic energy for He$^+$ ion at $n=2$

For He$^+$, it is a hydrogen-like ion with $Z=2$. Thus, $K_{He^+,n=2} = \frac{13.6(2)^2}{2^2} = 13.6$

4. Take the ratio

$\frac{K_{H,n=2}}{K_{He^+,n=2}} = \frac{13.6/4}{13.6} = \frac{1}{4}$

5. Answer

Therefore, the required ratio is $\boxed{\frac{1}{4}}$ So, Option A is correct.

6. Comparison with stored answer

Stored correct answer: A

Our derived answer: A

They agree.