E = F * s
E = (m*a) * (c*t), where m - mass, a - acceleration, c - velocity of light, and t - time.
E = (m*c/t) * (c*t), where a = velocity/time = c/t.
E = mc^2.
Let E be relativistic energy and K be kinetic energy
E = 2(K) = mc^2, where k = (mc^2)/2.
v = c/2 = 0.5c, where v is velocity of a moving body with mass m.
From classical physics, the change in the kinetic energy of a system $\Delta E_k$ is the work $W$ of a constant force of magnitude $F$ on a point that moves a displacement $s$ in a straight line in a direction of a force, that is $W = \Delta E_k = F \cdot s$. So, your formula is correct in classical physics (see https://en.wikipedia.org/wiki/Work_(physics)).
For the special relativistic physics ($v \approx c$), we have the following formula (see formula (9.6) in Landau and Lifshitz. Classical field theory)
$$\frac{E^2}{c^2} = p^2 + m^2 c^2$$ where $p$ is the momentum, $m$ is the rest mass (mass at rest), and $c$ is the speed of light.
If $p = 0$, that is, the system does not move, the above equation becomes $E = m c^2$.
Wrong... In the presence of a force, displacement isn’t proportional to time, nor is the velocity constant, so the calculation is inconsistent.
Instead of wasting time with meaningless calculations, it would be better to study any textbook on classical mechanics. This isn’t a research topic since 1905...
Energy is proportional to mass times the square of the velocity of light just from dimensional analysis. It’s the prefactor that matters and that this relation, with the correct prefactor, holds in a particular frame, that matters.
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