Kalman filter - question about matrices
According to the scenario you formulated, you only have measurement noise and no input noise. You are correct that you might need a state space model. For example,

$$
\begin{bmatrix}\dot{x} \\ \ddot{x}\end{bmatrix} = \begin{bmatrix} 0 &amp; 1 \\ 0 &amp; -\frac{b}{m}\end{bmatrix} \begin{bmatrix}x \\ \dot{x}\end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} u + \begin{bmatrix} 0 \\ 1 \end{bmatrix} v,
$$

$$
\begin{bmatrix}y_1 \\ y_2 \end{bmatrix} = \begin{bmatrix} 1 &amp; 0 \\ 0 &amp; 1 \end{bmatrix} \begin{bmatrix}x \\ \dot{x}\end{bmatrix} + \begin{bmatrix}w_1 \\ w_2\end{bmatrix},
$$

with $u$ always equal to $g$, $v$ the input noise (which assumed to be zero), $y_1$ the measured position, $y_2$ the measured velocity and $w_1$ and $w_2$ zero mean Gaussian noise.

Since we did not take into account input noise, therefore you can also view it as a zero mean Gaussian noise with also zero (standard) deviation. When calculating the covariance matrices you can take the variance (standard deviation squared) on the diagonal, if all random variables are independent (which I assume $w_1$ and $w_2$ are). This would yield,

$$
Q = \mathbb{E}(v^2) = \begin{bmatrix}0 \end{bmatrix},
$$

$$
R = \mathbb{E}\left([w_1\ w_2]^T [w_1\ w_2]\right)= \begin{bmatrix}225 &amp; 0 \\ 0 &amp; 25 \end{bmatrix}.
$$

But depending on whether you want to implement a Kalman filter in continues or discrete time, you might first have to discretize the state space model.