A rocket is fired straight up, burning fuel at the

Chapter , Problem 16

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A rocket is fired straight up, burning fuel at the constant rate of b kilograms per second. Let v vstd be the velocity of the rocket at time t and suppose that the velocity u of the exhaust gas is constant. Let M Mstd be the mass of the rocket at time t and note that M decreases as the fuel burns. If we neglect air resistance, it follows from Newtons Second Law that F M dv dt 2 ub where the force F 2Mt. Thus M dv dt 2 ub 2Mt Let M1 be the mass of the rocket without fuel, M2 the initial mass of the fuel, and M0 M1 1 M2. Then, until the fuel runs out at time t M2yb, the mass is M M0 2 bt. (a) Substitute M M0 2 bt into Equation 1 and solve the resulting equation for v. Use the initial condition vs0d 0 to evaluate the constant. (b) Determine the velocity of the rocket at time t M2yb. This is called the burnout velocity. (c) Determine the height of the rocket y ystd at the burnout time. (d) Find the height of the rocket at any time t.