Maximum Achievable Velocity Change

The rocket equation gives the ratio of the initial mass to the final mass given a velocity change and an exhaust velocity.

$$\frac{m_i}{m_f} = e^\frac{\Delta V}{V_e}$$

This seems to say that given enough fuel we could get an infinite velocity change! To see what the maximum possible velocity change could be we need to account for the structural fraction. The structural fraction multiplied by the mass of fuel gives the mass of the structure needed to support and contain the fuel. The rocket equation now is as follows

$$\frac{m_h + (1+f)m_p}{m_h + fm_p} = e^\frac{\Delta V}{V_e}$$

where $m_p$ is the mass of propellant, $f$ is the structural fraction, and $m_h$ is the mass of all other hardware. If we let the mass of propellant go to infinity, and solve for the velocity change, we get:

$$\frac{\Delta V}{V_e} = \log{\frac{1+f}{f}}$$

The following plot shows the ratio of velocity change to exhaust velocity for a range of structural fractions.

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