A 540-kg spacecraft is mounted on top of a rocket with a mass of 19 Mg, including 17.8 Mg of fue

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A 540-kg spacecraft is mounted on top of a rocket with a...

A $540-\mathrm{kg}$ spacecraft is mounted on top of a rocket with a mass of $19 \mathrm{Mg},$ including $17.8 \mathrm{Mg}$ of fuel. Knowing that the fuel is consumed at a rate of $225 \mathrm{kg} / \mathrm{s}$ and ejected with a relative velocity of $3600 \mathrm{m} / \mathrm{s}$, determine the maximum speed imparted to the spacecraft if the rocket is fired vertically from the ground.

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Key Concepts

Tsiolkovsky Rocket Equation

This fundamental relation in rocketry connects the change in velocity a rocket can achieve (delta-v) with the effective exhaust velocity and the logarithm of the mass ratio (initial mass divided by final mass). It encapsulates how efficiently mass is used to generate thrust and is derived from the principle of conservation of momentum for a system with varying mass.

Conservation of Momentum

This principle states that in the absence of external forces, the total momentum of a system remains constant. For rockets, as fuel is ejected in one direction, momentum is conserved by imparting an equal and opposite momentum to the rocket, providing the basis for its acceleration during flight.

Variable Mass Systems

Rockets are classic examples of variable mass systems because their mass decreases as fuel is burned and expelled. This characteristic necessitates special treatment in dynamics, since the standard equations for constant mass do not apply. Instead, one must account for how continuous mass loss influences the acceleration and velocity of the rocket over time.

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