Answer to Question #71480 in Classical Mechanics for Amy

Question

Multistage​ ​rocket​ ​in​ ​free​ ​space​ ​(no​ ​air,​ ​no​ ​gravity)​. Starting at rest with an initial total mass m0, and an assumed constant exhaust speed u with respect to the ship,
(a) what is the final velocity v of the ship with respect to the ground in a two-stage process where after stage 1 mass m1 of fuel has been burnt and then mass mdrop is jettisoned to prepare for stage 2 in which a total amount of fuel m2 is burnt?​ ​[In​ ​part​ ​(a)​ ​derive​ ​a​ ​formula​ ​for​ ​v​ ​in​ ​terms​ ​of​ ​u​ ​and​ ​arbitrary​ ​masses]
(b) What is the percentage increase or decrease of the final velocity v of the ship in this
two-stage process compared to a single-stage rocket with the same exhaust speed u and initial
total mass m0, but where just m1 + m2 of fuel is burnt? [In​ ​part​ ​(b)​ ​derive​ ​a​ ​number​ ​for​ ​v’s
percentage​ ​change,​ ​and​ ​say​ ​whether​ ​it​ ​increases​ ​or​ ​decreases​; like the Space Shuttle, use
mdrop = 2.5 ⨉ 105 kg, m2 = 2 mdrop , m1 = 2 m2, and m0 = 2 m1 = 2 ⨉ 106 kg]