# How to calculate the thrust of a rocket with relativistic exhaust

The general thrust equation is $F = \frac{dm}{dt}\cdot v$, where $m$ is propellant's mass and $v$ is the exhaust velocity, is the equation right? What if the propellant is highly relativistic? One gram of propellant expelled at $290,000,000\ \text{m/s}$ should produce $290,000\ \text{N}$, according to the above equation. However, this doesn't take relativistic change into account.

What I want to know is how to calculate thrust when relativistic exhaust is involved.

• Technically, the general thrust equation is $F=\dot{m}_ev_e+(p_e-p_0)A_e$ where $p_i$ are the exhaust & ambient pressures and $A_e$ the cross-sectional area of the nozzle. You've cited for the (ideal) case of $p_e=p_0$. – Kyle Kanos Jan 2 '16 at 15:02
• Just as an aside: the $I_{sp}$ of that rocket motor would be around 29.4 million seconds (almost a whole year)! – pr1268 Apr 16 '17 at 13:15

$$p = \gamma mv$$
where $\gamma$ is the Lorentz factor and $m$ is the rest mass of the body. So basically just multiply your non-relativistic calculation by $\gamma$.