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I am familiar with using Tsiolkovysky's Rocket Equation, $$\Delta v=v_{e} \ln \left(\frac{m_{i}}{m_{f}}\right)$$ However, after trying to work out an exhaust velocity needed in a hypothetical situation, I achieved a result higher than the speed of light. Obviously this is nonsense and a relativistic version of this equation is needed to handle these higher speeds. However after some searching I cant seem to find one.

If anyone could show me how to derive a version of this equation capable of dealing with relativistic exhaust velocities that would be much appreciated.

I am familiar with using Tsiolkovysky's Rocket Equation, $$\Delta v=v_{e} \ln \left(\frac{m_{i}}{m_{f}}\right)$$ However, after trying to work out an exhaust velocity needed in a hypothetical situation, I achieved a result higher than the speed of light. Obviously this is nonsense and a relativistic version of this equation is needed to handle these higher speeds. However after some searching I cant seem to find one.

If anyone could show me how to derive a version of this equation capable of dealing with relativistic exhaust velocities that would be much appreciated.

I am familiar with using Tsiolkovysky's Rocket Equation, $$\Delta v=v_{e} \ln \left(\frac{m_{i}}{m_{f}}\right)$$ However, after trying to work out an exhaust velocity needed in a hypothetical situation, I achieved a result higher than the speed of light. Obviously this is nonsense and a relativistic version of this equation is needed to handle these higher speeds. However after some searching I cant seem to find one.

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Relativstic Relativistic Rocket Equation

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Relativstic Rocket Equation

I am familiar with using Tsiolkovysky's Rocket Equation, $$\Delta v=v_{e} \ln \left(\frac{m_{i}}{m_{f}}\right)$$ However, after trying to work out an exhaust velocity needed in a hypothetical situation, I achieved a result higher than the speed of light. Obviously this is nonsense and a relativistic version of this equation is needed to handle these higher speeds. However after some searching I cant seem to find one.

If anyone could show me how to derive a version of this equation capable of dealing with relativistic exhaust velocities that would be much appreciated.