How much fuel is necessary for a rocket ship to complete acceleration from 100 m/s to 200 m/s with an exhaust gas speed of 1,000 m/s?

To determine the amount of fuel required for a rocket to accelerate from 100 m/s to 200 m/s, we can apply the Tsiolkovsky rocket equation, which is given as:

[

\Delta v = v_e \ln\left(\frac{m_0}{m_f}\right)

]

where:

• (\Delta v) is the change in velocity (200 m/s - 100 m/s = 100 m/s),

• (v_e) is the exhaust speed (1,000 m/s),

• (m_0) is the initial total mass of the rocket (including fuel),

• (m_f) is the final mass of the rocket (after fuel has been burned).

Rearranging the equation allows us to solve for the mass ratio (\frac{m_0}{m_f}):

[

\frac{m_0}{m_f} = e^{\frac{\Delta v}{v_e}}

]

Substituting in the known values:

[

\frac{m_0}{m_f} = e^{\frac{100 , \text{m/s}}{1000 , \text{m/s}}} =