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.
There is some little historical reference.

The Dirac equation has two linearly solutions which are the eigenstates of Dirac hamiltonian: first refers to the positive values of energy while the second refers to the negative ones. In classical field theory we can violate this problem by setting the trivial initial conditions $\psi (\mathbf r , t) $ for negative values, but in quantum theory it doesn't predict the absense of "negative" states, because there is possible discrete transitions between "positive" and "negative" states. The problem of negative energies is very important, because the absense of restrictions on the negative energy states promotes things like perpetuum mobile. It is experimentally unobserved. 

Dirac phenomenologically solved this problem by adding the Dirac sea conceprion to the free field theory. This gives the infinite number of bonded particles with negative energies, which makes possible the absence of free solutions with negative energies due to Pauli principle. But simultaneously it determines the processes of annihilating or processing particles. For example, if there is the idle state in the vacuum with negative energy $-E$, the free electron with energy $2E$ can take this state with emission of energy $2E$ (it is equal to annihilation of particles). So there is possible of decreasing of total number and charge of free particles.

But there's nothing critical for interpreting the Dirac theory as one-particle, because the bonded electrons can't interact between each other: interaction means transition from one bonded state to another one, but this is impossible because of the Pauli principle. So in the case of free Dirac particles this infinite number is unobservable due to isotropy and homogeneity of the space-time, and even after assuming the Dirac sea conception you can still interpret the Dirac equation as single-particle one. But lets introduce some external field. It can interact with the Dirac sea and pull out the particles from it. So the Dirac theory stops being one-particle one when the external field may give ene energy more than $2mc^2$.

All of these thinking leaded to the quantum field theory. All the ideas and problems which I described are still present in quantization formalizm (infinite vacuum energy, creation and destruction operators, scattering processes with pairs creation in an external field etc.), but it is important to distinguish the reasons and consequences. In our world there aren't free particles with negative energies - this is a reason why we introduce the quantum field theory.