Question: I'm trying to figure out which object would have the largest time of flight when launched from a slingshot. To do this, I need the initial velocity. How can I calculate this if I know the mass and diameter of the object? The slingshot always has the same force. I've seen a number of formulas, but none of them take into account the mass and size, which would matter in this case (a 10kg, 10cm rock will not go as fast or as long as a 1kg, 1cm rock).

Answer: 
Your problem is not related only with the Dirac equation, but with whatever field equation you have. Your question is about the interpretation of the field as a wave function of one particle. There are two concepts that I would like to show here that might answer your question: fields and wave functions (they are different things).

* A field is an object $\phi\_{\alpha} (x)$, where $x$ behaves as a event in space time and $\alpha$ is an index that carry a [finite representation of Lorentz Symmetry](https://en.m.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group#Finite-dimensional_representations). They obey some equations that related the $\alpha$ indices and the $x$, but this is irrelevant here. What is relevant is that they also obey the Klein Gordon equation, which is given by
$$
\partial^2\phi\_{\alpha}= \pm m^2\phi\_{\alpha},
$$
for the metric signature ($\pm,\mp, \cdots,\mp $).
* A wave function is a representation of a quantum state in a particular bases.
$$
\psi (x)=\langle x|\psi\rangle
$$
where $|x\rangle$ behave as a event (in non-relativistic quantum mechanics this is $(t,\vec {x}) $). Turns out that this state $|x\rangle$ in relativistic quantum mechanics is:
$$
|x\rangle=\int\frac{d^3p}{(2\pi)^{3/2}}\sqrt{\frac{k^0}{p^0}}e^{-ipx}|p\rangle
$$
You can prove yourself as an exercise. Note that there is a square root, which prevents this integral to be a simple Fourier transform of $|p\rangle$, the eigenvector of P (generator of translations).
The set of states $|x\rangle$ does not constitute an orthogonal basis since
$$
\langle x|y \rangle \neq 0
$$ 
and the square root in the formula above is the responsible for that. So, if we work with a wave function $\langle x|\psi\rangle$ in a relativistic QM we are dealing with am over-complete basis representation.

Now if we upgrade our language and move to the [second-quantization](https://en.wikipedia.org/wiki/Second_quantization) framework we have 
$$
|x\rangle=a^{\dagger}\_x|0\rangle
$$
Defining linear combinations of creation and annihilation operators of the form:
$$
\phi(x)=Aa\_x+Ba\_x^{\dagger}
 $$
such that $[\phi(x),\phi^{\dagger}(y)]= 0$ at space like separations is a useful construction to model local interactions of relativistic QM. This leads to the [Quantum Field Theory](https://en.wikipedia.org/wiki/Quantum_field_theory) enterprise.
The field is not interpreted as a wave function but as an operator $\hat{\psi}$ which creates/annihilates particles. This quantisation procedure is done by expanding the field in its Fourier components which depend on the momentum, spin, etc, $a\_s(p)$, $a\_s^\dagger(p)$, $b\_s(p)$, $b\_s^\dagger(p)$ (the fact that these $\hat{\psi}$ operators are not self conjugate requires the introduction of these two kinds) and interpreting them as creation and annihilation operators of particles/antiparticles with such momentum & spin. Then after imposing the right (anti)commutation relations for this operators and defining the vacuum state, $|0\rangle$ as the state annihilated by al the $a$s, we build the single and multi particle states as the $a^\dagger$s acting on the vacuum, just as we do with the simple harmonic oscillator. For instance
$$
|p\_1,s\_1;p\_2,s\_2\rangle = a\_{s\_1}^\dagger(p\_1) a\_{s\_2}^\dagger(p\_2) |0\rangle
$$
(I'm ignoring normalisation in case someone asks) And the same with the $b^\dagger$s for antiparticles.

Now the Dirac equation gives the evolution of such operators and the Hamiltonian itself, being a combination of this operators, acting on those states give positive energies. 

In case you wonder states of definite position are constructed with the field operators acting on the vacuum, so for instance the wave function of a definite momentum and spin state would be something like

$$
\psi(x) = \langle x | p,s\rangle = \langle 0 |\hat{\psi}\,a\_s^\dagger(p)|0\rangle
$$
again ignoring normalisation.