Estimating velocity from known position and acceleration
You would not make a constant-acceleration assumption in this case. You would typically do that if you had no means of measuring the system's acceleration, but you say that it is observable for your case. The most obvious way to model this system would be using the state vector

$$
\mathbf{x_k} = \left [ 
\begin{array}{c}x_k \\
\dot{x}_k \\
\ddot{x}_k
\end{array} \right ] = \left [ 
\begin{array}{c}x_k \\
v_k \\
a_k
\end{array} \right ]
$$

where $x_k$ is the position, $v_k$ is the velocity, and $a_k$ is the acceleration of the system at time instant $k$. Since you say that you can measure the position and acceleration of the system, the measurement vector $\mathbf{z_k}$ would be:

$$
\mathbf{z_k} = \mathbf{Hx_k} + \mathbf{v_k}
$$

with

$$
\mathbf{H} = \left[ 
\begin{array}{cc} 
1 &amp;&amp; 0 &amp;&amp; 0 \\
0 &amp;&amp; 0 &amp;&amp; 1
\end{array}
\right]
$$

resulting in the measurement model:

$$
\mathbf{z_k} = \left[
\begin{array}{c}
x_k \\
a_k
\end{array}
\right] + \mathbf{v_k}
$$

where $\mathbf{v_k}$ is the (Gaussian) noise inherent in the measurement. Now, unless you are adding some known force to the system that would affect its dynamics, then the input vector $\mathbf{u_k} = \mathbf{0}$. See this recent question for more detailed discussion of the various terms in the Kalman model, which you need to know, and which you don't.

Edit: Regarding the below comment on how to handle the state transition matrix for the acceleration term: there are a couple different ways to handle that. You could assume a zero-jerk (jerk is the time derivative of acceleration) model; this is equivalent to assuming that $a_{k+1}=a_k$. Or, you could add a fourth element to your state vector:

$$
\mathbf{x_k} = \left [ 
\begin{array}{c}x_k \\
\dot{x}_k \\
\ddot{x}_k \\
\dddot{x_k}
\end{array} \right ] = \left [ 
\begin{array}{c}x_k \\
v_k \\
a_k \\
j_k
\end{array} \right ]
$$

where $j_k$ is the system's jerk, or time rate of change of acceleration, at time instant $k$. Since you aren't measuring jerk directly, you would have a measurement matrix of:

$$
\mathbf{H} = \left[ 
\begin{array}{cc} 
1 &amp;&amp; 0 &amp;&amp; 0 &amp;&amp; 0\\
0 &amp;&amp; 0 &amp;&amp; 1 &amp;&amp; 0
\end{array}
\right]
$$

And the state transition matrix would become:

$$
\mathbf{F} = \left[ 
\begin{array}{cccc} 
1 &amp;&amp; dt &amp;&amp; \frac{1}{2}dt^2 &amp;&amp; \frac{1}{6} dt^3 \\
0 &amp;&amp; 1 &amp;&amp; dt &amp;&amp; \frac{1}{2}dt^2 \\
0 &amp;&amp; 0 &amp;&amp; 1 &amp;&amp; dt \\
0 &amp;&amp; 0 &amp;&amp; 0 &amp;&amp; 1
\end{array}
\right]
$$

This is a "constant-jerk" model (similar to constant-velocity or constant-acceleration models you might see in simple examples); your state transition matrix makes an implicit assumption that the jerk is constant for all values of k. This is not likely to be true. To handle that aspect of the problem, you would introduce a process noise term in the jerk component of the state transition equation.

Qualitatively, this allows you to express that you aren't sure how the jerk term is going to change from time step to time step, but that you expect the changes to be random with some Gaussian distribution. This is a tool often used to "tune" adaptive filter models like this one until you find a set of parameters that works well for your application.