The first stage of a Saturn V space vehicle consumes fuel...

The first stage of a Saturn $\mathrm{V}$ space vehicle consumes fuel and oxidizer at the rate of $1.50 \times 10^{4} \mathrm{~kg} / \mathrm{s}$, with an exhaust speed of $2.60 \times 10^{3} \mathrm{~m} / \mathrm{s}$. (a) Calculate the thrust produced by these engines. (b) Find the initial acceleration of the vehicle on the launch pad if its initial mass is $3.00 \times 10^{6} \mathrm{~kg} .[$ Hint : You must include the force of gravity to solve part (b).]

Key Concepts

Thrust

Thrust is the force produced by an engine due to mass being expelled at high velocity. It is calculated by multiplying the mass flow rate (the mass of fuel expelled per unit time) by the exhaust velocity. This concept originates from the conservation of momentum, where the momentum carried away by the expelled mass results in an equal and opposite reaction force on the vehicle.

Mass Flow Rate

Mass flow rate quantifies the amount of mass expelled per unit time by an engine. It is a critical parameter in rocket propulsion that determines how much mass is available to generate thrust through high-speed ejection, thereby influencing the overall force generated by the engine.

Exhaust Velocity

Exhaust velocity is the speed at which propellant exits the engine. In rocket propulsion, higher exhaust velocities generally lead to higher thrust for a given mass flow rate according to the relation thrust = mass flow rate × exhaust velocity. This parameter plays a central role in determining the efficiency and performance of rocket engines.

Newton’s Second Law

Newton’s Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (a = F/m). In the context of rocketry, this principle is used to determine the acceleration of a spacecraft by considering both the thrust generated by the engine and any opposing forces, such as gravity.

Gravitational Force

Gravitational force is the force exerted on an object due to gravity, calculated as the product of its mass and the acceleration due to gravity. When analyzing the motion of a rocket during launch, the gravitational force acts in opposition to the thrust, and its effect must be subtracted from the engine's thrust to determine the net force and thus the initial acceleration of the vehicle.

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