Randall D. Knight — Impulse and Momentum: Problem 85

Section 11.6 found an equation for \(v_{\text{max}}\) of a rocket fired in deep space. What is \(v_{\text{max}}\) for a rocket fired vertically from the surface of an airless planet with free-fall acceleration \(g\)? Referring to Section 11.6, you can write an equation for \(\Delta P_y\), the change of momentum in the vertical direction, in terms of \(dm\) and \(dv_y\). \(\Delta P_y\) is no longer zero because now gravity delivers an impulse. Rewrite the momentum equation by including the impulse due to gravity during the time \(dt\) during which the mass changes by \(dm\). Pay attention to signs! Your equation will have three differentials, but two are related through the fuel burn rate \(R\). Use this relationship—again pay attention to signs; \(m\) is decreasing—to write your equation in terms of \(dm\) and \(dv_y\). Then integrate to find a modified expression for \(v_{\text{max}}\) at the instant all the fuel has been burned. (a) What is \(v_{\text{max}}\) for a vertical launch from an airless planet? Your answer will be in terms of \(m_R\), the empty rocket mass; \(m_{F0}\), the initial fuel mass; \(v_{\text{ex}}\), the exhaust speed; \(R\), the fuel burn rate; and \(g\). (b) A rocket with a total mass of 330,000 kg when fully loaded burns all 280,000 kg of fuel in 250 s. The engines generate 4.1 MN of thrust. What is this rocket’s speed at the instant all the fuel has been burned if it is launched in deep space? If it is launched vertically from the earth?