A two-stage rocket is to be built capable of acceleration a100-kgm payload to a

Question

A two-stage rocket is to be built capable of acceleration a100-kgm payload to a velocity of 6000 m/s in free flight in emptyspace (no gravitational field). (In a two-stage rocket, the firststage is detached after exhausting its fuel, before the secondstage is fired.) Assume that the fuel used can reach an exhaustvelocity of 1500 m/s , and that structural requirements imply thatan empty rocket ( without fuel or payload) will weigh 10% as muchas the fuel it can carry. Find the optimum choice of masses for thetwo stages so that the total take-off weight is a minimum. Showthat it is impossible to build a single-stage rocket which will dothe job.

Explanation / Answer

I feel the data is insufficient. A two stage rocket. the velocity of a rocket is given by       V = Vo + uloge ( Mo /M ) - gt     Vo = initial velocity
    u = exhaust velocity

   ignoring the gravity, the equation

          V =Vo + u loge ( Mo/ M)    
for first stage
               V1 =Vo + u loge ( Mo/M)---------(1) for second stage       V2 = V1 +u loge (Mo / M)----------(2)      equation (2) given

   V2 = 6 km /s
   u = 1.5 km / s
loge ( Mo / M) = 2.3026log10 ( 100 / 10 )
        sub. andsimplfing
  
    V1 = 2.415 km/s    sub. the value of in equation (1)      2.415 = 0 + u loge (Mo/M)                            for first stage Vo = 0
     loge ( Mo / M) = 2.415 /1.5
                               =1.61
the weight of the first satge is----

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