For a rocket, find the ratio of its propellant mass to the initial mass for which burnout velocity of the rocket is equal to the escape velocity from earth.
For a rocket, find the ratio of its propellant mass to the initial mass for which burnout velocity of the rocket is equal to the escape velocity from earth. The rocket uses hydrogen-fluorine as propellants which has a specific impulse of 390 s.
Burnout velocity: It is the velocity of the rocket when propellants stop burning. It is the maximum velocity achieved by a rocket.
Escape velocity : It is the minimum velocity required for a body to escape from a gravitational center of attraction. Kinetic energy of the body is equal to the potential energy and in the absence of any frictional resistance the body will escape from the planet. The escape velocity from earth is \(11.2\, km/s\). Burnout velocity for a rocket is given by
\[{V_b} = {g_0}{I_{sp}}\ln \left( {\frac{{{M_i}}}{{{M_f}}}} \right)\]
\[ \Rightarrow 11.2 \times 1000 = \left( {9.8} \right)\left( {390} \right)\ln \left( {\frac{{{M_i}}}{{{M_f}}}} \right)\]
\[ \Rightarrow \left( {\frac{{{M_i}}}{{{M_f}}}} \right) = 18.74\]
Also,
\[{M_i} = \left( {{M_p} + {M_f}} \right)\]
\[ \Rightarrow {M_f} = \left( {{M_i} – {M_p}} \right)\]
\[ \Rightarrow \frac{{{M_i}}}{{{M_f}}} = \frac{{{M_i}}}{{{M_i} – {M_p}}} = \frac{1}{{1 – \left( {\frac{{{M_p}}}{{{M_i}}}} \right)}}\]
\[ \Rightarrow 18.74 = \frac{1}{{1 – \left( {\frac{{{M_p}}}{{{M_i}}}} \right)}}\]
\[ \Rightarrow 1 – \left( {\frac{{{M_p}}}{{{M_i}}}} \right) = \frac{1}{{18.74}}\]
\[ \Rightarrow \frac{{{M_p}}}{{{M_i}}} = 0.947\]
