Suppose that a rocket of mass M blasts off from a spacestation in free space at

Question

Suppose that a rocket of mass M blasts off from a spacestation in free space at time t=0. The rocket pushed forward byexhaust gases that rearward with constant speed c (relative torocket) and the rocket fuel is consumed at the constant "burn rate". Find maximum possible speed of the rocket. please show all steps to solve anyway possible....thx Suppose that a rocket of mass M blasts off from a spacestation in free space at time t=0. The rocket pushed forward byexhaust gases that rearward with constant speed c (relative torocket) and the rocket fuel is consumed at the constant "burn rate". Find maximum possible speed of the rocket. please show all steps to solve anyway possible....thx

Explanation / Answer

In the absence of any outside forcesNewton's third law holds and momentum is conserved. If we let p(t)denote the rocket's momentum at time t, v therocket's velocity, and m the rocket's mass thenp(t)=m v. The rocket is burning fuel which causesit to lose mass which is exhausted from the rocket at a velocity ofc. An instant after time t the following equationwould hold because v-c is the velocity of theejected fuel with respect to the direction of the rocket's traveland momentum is conserved:    p(t + dt) = (m + dm)(v +dv) - dm(v - c) = mv + mdv + cdm Subtracting the equationp(t)=m v from this equation we obtain dp = mdv + cdm. In the absence ofany external forces dp=0 because momentumis conserved, and dividing both sides by dt we are left with the equation of a thrusting rocket whichis simply:    m(dv/dt) =-c(dm/dt) Now if the rocket ejects mass as a constantrate - = dm/dt we can write m = M-t and theequation becomes:    (M - t)(dv/dt) =c Solving this equation with the initialcondition that v=0 at t=0 we have the solution:    v(t) = c log[M] - c log[M- t] Now the rocket certainly has mass inaddition to that of its fuel. Let's assume that after all the fuelis burned the rocket has a remaining mass of m. This means that M -t = m once all the fuel is burned. The maximum speed of therocket is then:    vmax = clog[M/m] This is a considerable limitation in thedesign of rockets. In a very extreme cases we might have M/m˜ 10 which greatly limits the maximum possiblespeed.    m(dv/dt) =-c(dm/dt) Now if the rocket ejects mass as a constantrate - = dm/dt we can write m = M-t and theequation becomes:    (M - t)(dv/dt) =c Solving this equation with the initialcondition that v=0 at t=0 we have the solution:    v(t) = c log[M] - c log[M- t] Now the rocket certainly has mass inaddition to that of its fuel. Let's assume that after all the fuelis burned the rocket has a remaining mass of m. This means that M -t = m once all the fuel is burned. The maximum speed of therocket is then:    vmax = clog[M/m] This is a considerable limitation in thedesign of rockets. In a very extreme cases we might have M/m˜ 10 which greatly limits the maximum possiblespeed.        

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