The rocket engines of a spacecraft are fired to increase its...

The rocket engines of a spacecraft are fired to increase its velocity
by $450 \mathrm{ft} / \mathrm{s}$. Knowing that $1200 \mathrm{lb}$ of fuel is ejected at a relative velocity of $5400 \mathrm{ft} / \mathrm{s}$, determine the weight of the spacecraft after the firing.

Key Concepts

Impulse-Momentum Theorem

The impulse-momentum theorem relates the change in momentum of an object to the impulse applied, which is the product of force and the time interval during which the force acts. In the case of a rocket, the impulse can be determined from the product of the mass of the expelled fuel and its relative velocity. This relationship is essential for quantifying the effect of fuel ejection on the spacecraft's velocity and mass.

Exhaust Velocity

Exhaust velocity is the speed at which fuel is expelled relative to the spacecraft. It is a critical factor in rocket propulsion because it determines how effectively the fuel's mass converts into momentum change for the spacecraft. A higher exhaust velocity generally means a more efficient propulsion system, impacting the calculations of velocity increments and the resulting weight of the spacecraft after fuel burn.

Conservation of Momentum

This concept states that in an isolated system (with no external forces acting), the total momentum remains constant. In the context of rocket propulsion, when fuel is expelled in one direction, the spacecraft gains momentum in the opposite direction. This principle is used to analyze the changes in the spacecraft's velocity as fuel is fired.

Rocket Propulsion

Rocket propulsion involves the acceleration of a vehicle through the expulsion of mass at high speed. The basic principle relies on the conservation of momentum, where the momentum change in the expelled fuel results in a reactive force on the spacecraft, leading to its acceleration. Understanding this process is crucial for calculating parameters such as the velocity increase and the remaining mass or weight of the spacecraft after fuel burn.

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