Question

A rocket in deep space has an empty mass of $150$ kg and exhausts the hot gases of burned fuel at $2500$ m/s. What mass of fuel is needed to reach a top speed of $4000$ m/s?

Short answer

The mass of fuel needed to reach the maximum speed is$593kg$.

Step-by-step solution

Step 1. Given information

Mass of rocket, $m_R=150kg$

speed of exhaust gas, $v_{ex}=2500m/s$

maximum speed,$v_{max}=4000m/s$

Step 2. Analyze

When the fuel is entirely burnt, the rocket exceeds the threshold speed. As a result, the entire quantity of the fuel has been used, and we have reached our goal. The rocket's maximum speed is calculated using an equation of the type

$$\begin{gathered} v_{\max }=v_{ex}\ln \left(\frac{m_R+m_{F0}}{m_R}\right) \\ \frac{v_{max}}{v_{ex}}=\ln \left(\frac{m_R+m_{F0}}{m_R}\right) \\ {e}^{\left(\frac{v_{\max }}{v_{ex}}\right)}=\left(\frac{m_R+m_{F0}}{m_R}\right) \\ {m}_{F0}=m_R\left(e^{v_{\max }/v_{ex}}-1\right) \end{gathered}$$

Step 3. Evaluate

Substitute and evaluate,

$$\begin{gathered} m_{F0}=m_R\left(e^{v_{\max }/v_{ex}}-1\right) \\ {m}_{F0}=150\left(e^{4000/2500}-1\right)=593kg \end{gathered}$$