- #1

tony873004

Science Advisor

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A sounding rocket launched from Earth’s surface is to achieve a final speed of 1000 m/s. If the exhaust speed of the spent fuel is 2000 m/s, what fraction of the rocket’s total mass at launch must be fuel? (Assume that the engine burns rapidly enough that you may ignore any effects due to Earth’s gravity during the burn.)

[tex]

\[

v_f =v_i +v_{ex} \ln \left[ {\frac{m_i }{m_f }} \right]

\]

\[

v_f -v_i =v_{ex} \ln \left[ {\frac{m_i }{m_f }} \right]

\]

\[

\ln \left[ {\frac{m_i }{m_f }} \right]=\frac{v_f -v_i }{v_{ex} }

\]

\[

\ln \left[ {\frac{m_i }{m_f }} \right]=\frac{1000m/s-0m/s}{2000m/s}

\]

\[

\ln \left[ {\frac{m_i }{m_f }} \right]=0.5

\]

\[

\ln ^{-1}\left[ {0.5} \right]=1.6487

\]

\[

\frac{m_i }{m_f }=1.67487

\]

\[

\frac{m_f }{m_i }=\frac{1}{1.67487}

\]

\[

\frac{m_f }{m_i }=0.60653

\]

[/tex]