Sorry for what might be a long post, I want to give background.
Initially I had regular Kalman filter, and the state model was defined by Newtonian kinematics, with initial position 0 and speed of 2. I was tracking position (x) and velocity (v), i.e. my state vector is $\begin{bmatrix} x & \dot x \\ \end{bmatrix}^T$:
$$x = x_0 + v_0t$$ $$v = v_0$$
This resulted in a State Transition Matrix: $$ \begin{bmatrix} 1 & \Delta t\\ 0 & 1\\ \end{bmatrix}$$
Now I am trying to implement Extended Kalman Filter. I have given system Acceleration of 2, so that equations go like this together with plugging in initial speed and acceleration:
$$x = x_0 + v_0t + \frac12at^2 \rightarrow x = 2t + t^2$$
$$v = v_0 + at \rightarrow v = 2 + 2t$$
Now I need to find Jacobian Matrix with respect to my state vector and I understand what it is, however, I do not understand, how do I find my State Transition Matrix, if equations that I have are expressed in terms of time and not in terms of the state variables. I assume that first line in State Transition Matrix remains the same, since position changes in same way, i.e: $$position = previous\,position + \Delta time * speed\, over\, that \, time\, period$$ It is the speed that is changing. But I don't know how to define it in my State Transition Matrix. From what I understand, my STM will be different every epoch, am I right? As I said, I know that I need to find Jacobian and know what it is, but I don't know how to find it in this particular case.
Thank you.
I suggest reading a few tutorials on Kalman filter or "Optimal State Estimation" by Dan Simon. I think you are misunderstanding the Kalman filter, and your formulation doesn't make much sense. However, I will try to clarify some of the misunderstandings.
In a standard Kalman filter, the process model and observation models are linear i.e., $$ \mathbf{x}_k = \mathbf{A}_k \mathbf{x}_{k-1} + \mathbf{B}_k \mathbf{u}_k$$ $$ \mathbf{y}_k = \mathbf{C}_k \mathbf{x}_k$$ where $\mathbf{A}$ is the process model (or state transition matrix), $\mathbf{B}$ is the control input model, and $\mathbf{C}_k$ is the observation model (noting that I did not include the noise terms). The state and output are $\mathbf{x}_k$ and $\mathbf{y}_k$, respectively. I suggest trying to write the matrices $\mathbf{A}$ and $\mathbf{B}$ for your system.
For an extended Kalman filter, the process and observation models are nonlinear i.e.,
$$ \mathbf{x}_k = \mathbf{f}(\mathbf{x}_{k-1},\mathbf{u}_{k-1}) $$ $$ \mathbf{y}_k = \mathbf{h} (\mathbf{x}_k)$$
where $\mathbf{f}$ and $\mathbf{h}$ are vector valued functions (noting I did not include the noise terms here either). Now, in the nonlinear case, you do not have a state transition matrix because the system is nonlinear. If you do not understand why this is the case, you should start here, then as I said, look up some tutorials or the Dan Simon book. I hope this is helpful to you, but I prefer not to write entire KF tutorial here.
Now, to answer your question... the Jacobian $\mathbf{F}$ in general is given by $$ \mathbf{F}_k = \frac{\partial f}{\partial \mathbf{x}} \Biggr|_{\hat{\mathbf{x}}_{k-1|k-1}} $$
where $\hat{\mathbf{x}}_{k-1|k-1}$ is your previous estimate of your state and $f$ is the process model. You provided the process model, so you can simply compute the Jacobian from this formula. If you need help with this, I suggest looking up how to compute partial derivatives. Then, after you are good on that part, try to apply this formula. Best of luck!