# Force Drag and impact kinetic energy on relativistic spacecraft

Assuming a relativistic rocket travelling at 0.95 times the speed of light (c), what would be the drag force on the cross-section area $(\pi500^2)$ of the ship facing the direction of travel assuming here that drag coefficient is 0.25. The equation for force drag in classical mechanics is: $$F_D = \frac12\ pv^2C_DA$$ where $p$ is density of interstellar space; which should be about 2 million protons per $m^3$.

However I am not sure if in relativistic mechanics it is as such: $$F_D = pv^2\gamma^2$$ where $\gamma$ is the lorentz factor (gamma factor).

Both of these equations will give different results.

Furthermore, what is the kinetic energy when matter within the vacuum of space impacts on cross-section area $(\pi500^2)$ of the ship facing the direction of travel?

The equation is as follows: $$K_E = (\gamma-1)pc^2$$

However the results from these equations are about 26.6 N, 0.0028 N and $6.6e^-4$ J respectively. The latter is not much! However in many articles on the web and as per certain experts, the KE should have been greater and as explosive as nuclear bombs?! Where am I wrong?

• You might find the following useful: https://what-if.xkcd.com/1/. I think the drag force will not be anywhere as simple as just adding a Lorentz factor. Mar 29, 2017 at 13:32