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 Mentor The total energy is given by $$\frac{mc^2}{\sqrt{1-v^2/c^2}}$$ It is also given by $$mc^2+w$$ Set those two expressions equal and solve for v which gives: $$v=\pm\frac{c \sqrt{w \left(2 c^2 m+w\right)}}{c^2 m+w}$$