Two identical point masses, $P$ and $Q$, are suspended from two separate massless springs with spring constants $k_1$ and $k_2$, respectively. These masses oscillate vertically, and it is given that their maximum speeds are the same. We need to determine the ratio of the amplitude $A_Q$ of mass $Q$ to the amplitude $A_P$ of mass $P$.

The maximum velocity $ V $ for an oscillating mass is given by the equation $ V = A \omega $, where $ A $ is the amplitude and $ \omega $ is the angular frequency.

Given that the maximum velocities of $P$ and $Q$ are the same:

$ v_P = v_Q $

This implies:

$ A_P \omega_P = A_Q \omega_Q $

From this, the ratio of the amplitudes can be expressed as:

$ \frac{A_Q}{A_P} = \frac{\omega_P}{\omega_Q} $

The angular frequency $ \omega $ of a mass-spring system is given by:

$ \omega = \sqrt{\frac{k}{m}} $

Thus, for the two masses:

$ \omega_P = \sqrt{\frac{k_1}{m}} \quad \text{and} \quad \omega_Q = \sqrt{\frac{k_2}{m}} $

Substituting these into the ratio equation, we get:

$ \frac{A_Q}{A_P} = \frac{\sqrt{\frac{k_1}{m}}}{\sqrt{\frac{k_2}{m}}} = \sqrt{\frac{k_1}{k_2}} $

Hence, the ratio of the amplitude of mass $Q$ to the amplitude of mass $P$ is:

$ \sqrt{\frac{k_1}{k_2}} $