A rocket for use in deep space is to be capable of boosting a total load (payloa

Question

A rocket for use in deep space is to be capable of boosting a total load (payload plus rocket frame and engine) of 3.00 metric tons to a speed of 10,000 m/s.

(a) It has an engine and fuel designed to produce an exhaust speed of 2,000 m/s. How much fuel plus oxidizer is required?

(b) If a different fuel and engine design could give an exhaust speed of 5,000 m/s, what amount of fuel and oxidizer would be required for the same task?

(c) Noting that the exhaust speed in part (b) is 2.50 times higher than that in part (a), explain why the required fuel mass is not simply smaller by a factor of 2.50.

Explanation / Answer

assume that there is a very small amount of fuel and oxidizer mass dm in a period of dt be pushed out from the rocket. we have -dm*u=m*dv. where u be the exhaust speed m be the total mass of the system at that time and v be the velocity of the rocket. so dm/m*u=dv. so ln(m1/m0)*u=-v. so ln(m0/m1)*u=v. m0=m1+m_fuel. m1 be the mass of the rocket without fuel. so m0/m1=e^(v/u) m_fuel=m1*(e^(v/u)-1) ------ a). m_fuel = 442(metric tons) b). m_fuel = 19.2(metric tons). ----- c) because the fuel is pushed out slowly and not immediately.

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