How to use Kalman filter for altitude prediction based on barometer data?
You might get good results when you consider the acceleration as an input $u$, so the model could then be written as

$$
x[k+1] = 
\begin{bmatrix}
1 &amp; \Delta t \\
0 &amp; 1
\end{bmatrix} x[k] + 
\begin{bmatrix}
\frac{\Delta t^2}{2} \\ \Delta t
\end{bmatrix} u[k] + w[k]
$$

$$
y[k] = 
\begin{bmatrix}
1 &amp; 0
\end{bmatrix} x[k] + v[k]
$$

where $x[k]$ a vector with the position and velocity and $w[k],v[k]$ zero mean Gaussian white noise. Here $v[k]$ has a covariance $R$ equal to the variance of the barometer and $w[k]$ probably has the following covariance $Q$

$$
Q = \sigma_u^2\,
\begin{bmatrix}
\frac{\Delta t^2}{2} \\ \Delta t
\end{bmatrix}
\begin{bmatrix}
\frac{\Delta t^2}{2} \\ \Delta t
\end{bmatrix}^\top
= \sigma_u^2\,
\begin{bmatrix}
\frac{\Delta t^4}{4} &amp; \frac{\Delta t^3}{2} \\
\frac{\Delta t^3}{2} &amp; \Delta t^2
\end{bmatrix}
$$

where $\sigma_u^2$ the variance of the measured acceleration (this assumes that there are not disturbance forces acting on the system).