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Two missiles A and B powered by solid rocket motors have identical specificimpulse, liftoff mass of 5600 kg each, and burn durations of30At = sand70Bt = s, respectively. The propellant mass flow rates,Am andBm , formissiles A and B, respectively, are given by120 kg/s, 0 3070 kg/s, 0 70ABttmm=  =  Neglecting gravity and aerodynamic forces, the relationship between the finalvelocitiesAV andBV of missiles A and B, respectively, is given by(A)4.1 BAV V=(B)BAV V=(C)0.5 BAV V=(D)0.7 BAV V=

Question

Two missiles A and B powered by solid rocket motors have identical specificimpulse, liftoff mass of 5600 kg each, and burn durations of30At = sand70Bt = s, respectively. The propellant mass flow rates,Am andBm , formissiles A and B, respectively, are given by120 kg/s, 0 3070 kg/s, 0 70ABttmm=  =  Neglecting gravity and aerodynamic forces, the relationship between the finalvelocitiesAV andBV of missiles A and B, respectively, is given by(A)4.1 BAV V=(B)BAV V=(C)0.5 BAV V=(D)0.7 BAV V=

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Solution

The relationship between the final velocities AV and BV of missiles A and B, respectively, can be determined by considering the specific impulse and burn durations of the missiles.

Given that the specific impulse is identical for both missiles A and B, we can assume that the exhaust velocities are the same. Therefore, the change in velocity for each missile can be calculated using the equation:

ΔV = Isp * g0 * ln(m0/mf)

where ΔV is the change in velocity, Isp is the specific impulse, g0 is the standard gravity, m0 is the initial mass, and mf is the final mass.

Since the liftoff mass is the same for both missiles (5600 kg), we can calculate the final mass for each missile using the propellant mass flow rate and burn duration:

mAf = m0 - Am * At mBf = m0 - Bm * Bt

Substituting the given values, we have:

mAf = 5600 kg - 120 kg/s * 30 s = 5600 kg - 3600 kg = 2000 kg mBf = 5600 kg - 3070 kg/s * 70 s = 5600 kg - 214900 kg = -209300 kg

Since the final mass for missile B is negative, it means that the propellant mass flow rate is greater than the initial mass, which is not physically possible. Therefore, we can conclude that missile B will not have a final velocity.

Hence, the relationship between the final velocities AV and BV of missiles A and B, respectively, is given by:

AV = 4.1 * BV

Therefore, the correct answer is (A) 4.1 BAV V=.

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