For any free particle in special relativity:
$$E \vec{v} = c^2 \vec{p} \\ E^2 - \left( c \vec{p} \right)^2 = \left( m c^2 \right) ^2$$
So $$\vec{v}, \, \vec{p} \, \textrm{and} \, E$$ depend on your choice of coordinate system, while $$c \, \textrm{and} \, m$$ do not.
If $$m > 0$$ then we may, motivated by the trigonometry-like second equation, define $$\rho = \cosh^{\tiny -1} \frac{E}{mc^2}$$ and soon come to the conclusion:
$$ E = m c^2 \, \cosh \rho \\ \vec{p} = m c \, \sinh \rho \; \hat{v} \\ \vec{v} = c \, \tanh \rho \; \hat{v}$$
Since $$ \cosh \, \tanh^{\tiny -1} x = \frac{1}{\sqrt{1 - x^2}} \; \textrm{and} \; \sinh \, \tanh^{\tiny -1} x = \frac{x}{\sqrt{1 - x^2}}$$ these trigonometric identities then allow us to fairly state everything in terms of one free parameter: $$\vec{v}$$
$$ E = \frac{m c^2}{\sqrt{1 - \left(\frac{\vec{v}}{c}\right)^2}} \\ \vec{p} = \frac{m \vec{v}}{\sqrt{1 - \left(\frac{\vec{v}}{c}\right)^2}}$$
If $$m > 0$$ then there exists an inertial coordinate system where $$\vec{v} = 0, \, \vec{p} = 0 , \, \textrm{and} E = mc^2$$ thus we are justified in talking about rest energy, $$E_0 \equiv m c^2$$, and the additional energy associated with movement relative to the chosen inertial coordinate system, or kinetic energy for short, $$\textrm{KE} \equiv E - E_0 = \sqrt{ \left(m c^2 \right) ^2 \, + \, \left( c \vec{p} \right)^2 } - mc^2 = mc^2 \left( \frac{1}{\sqrt{1 - \left(\frac{\vec{v}}{c}\right)^2}} - 1 \right)$$
Therefore, since the velocity of your bowling ball in space depends on a choice of inertial coordinate systems (because only in inertial coordinate systems does a free particle have a constant velocity, by Newton's law of inertia) but its Energy, Kinetic Energy and momentum also depend on this choice in the precise mathematical relationship explained above.
//Edit:
And it is because of precision experiments that we have confidence that both special relativity and the above relationships between velocity, momentum and energy hold to high precision. We see relativity supported over Newtonian predictions in precision experiments with high-speed phenomena as diverse as the speed of light in moving water (Fizeau, 1859) or the relation between energy and velocity for electrons (Bertozzi, 1964; Fan, unpublished). Fan, for example, was a relativity denier and a terrible experimentalist, but even with gross systemic effects his observations agreed in detail with the Relativity that the kinetic energy of a body increases with velocity at a rate higher than Newton predicted.
If you graph velocity squared divided by the speed of light squared on the horizontal axis and kinetic energy divided by the Newtonian prediction of kinetic energy on the vertical axis, then relativity predicts a upward curving line with slope of 3/8 near v=0. Newton predicts zero.
Fan, in his paper, predicts -1/8. So the upward slope from Bertozzi (and Fan) got causes us to favor Relativity over the theories of Fan and Newton.
Theory:
Fan's Conjecture (green) versus Imprecise Reflections from Reality (Fan's experimental data, blue):
All other precision observation also supports special relativity in its domain of applicability (where gravity effects can be ignored).
