Question

An ion thruster mounted in a satellite uses electric forces to eject xenon ions with a speed of \(21.45 \mathrm{~km} / \mathrm{s}\). The ion thruster produces a thrust of \(1.187 \cdot 10^{-2} \mathrm{~N}\). What is the rate of fuel consumption of the thruster?

Short answer

Answer: The rate of fuel consumption is \(5.533 \times 10^{-7} \ \text{kg/s}\).

Step-by-step solution

List the given information

We are given the following information: - Speed of xenon ions \(v = 21.45 \ \text{km/s} = 21,\!450 \ \text{m/s}\) - Thrust of ion thruster \(T = 0.01187 \ \text{N}\)

Use Newton's second law to find the mass flow rate

According to Newton's second law, force equals mass times acceleration (\(F = ma\)). However, in our case, force is the thrust produced by the ion thruster and "acceleration" refers to the speed of the xenon ions multiplied by the mass flow rate (rate of fuel consumption). Therefore, we have: \(T = \dot{m}v\) Where \(\dot{m}\) is the mass flow rate (rate of fuel consumption).

Solve for the mass flow rate

Now we need to solve for the mass flow rate \(\dot{m}\) by using the given values for thrust and speed of xenon ions: \(\dot{m} = \frac{T}{v} = \frac{0.01187 \ \text{N}}{21,\!450 \ \text{m/s}}\)

Calculate the mass flow rate

Plugging in the given values, we get the mass flow rate: \(\dot{m} = \frac{0.01187 \ \text{N}}{21,\!450 \ \text{m/s}} = 5.533 \times 10^{-7} \ \text{kg/s}\)

Write the final answer

The rate of fuel consumption of the ion thruster is \(5.533 \times 10^{-7} \ \text{kg/s}\).

Key concepts

Newton's Second Law

Newton's second law is a fundamental principle in physics that describes how the velocity of an object changes when it is subjected to an external force. The law states that the force applied on an object is equal to the mass of the object multiplied by the acceleration it experiences, often written as the equation:
\[\begin{equation} F = ma \[10pt] \[10pt]\end{equation}\]
In the context of ion thruster fuel consumption, Newton's second law helps us understand the relationship between the thrust produced by the ion thruster (the force) and how the mass of the fuel, being expelled at a certain speed (the acceleration), relates to this force. Essentially, the ion thruster's efficiency and its capability to propel a spacecraft are derived from this basic principle.

Mass Flow Rate

Mass flow rate, often denoted by \(\dot{m}\), is a measure of the mass of substance passing through a given surface per unit time. It is a critical component in the equation for thrust in propulsion systems like ion thrusters, defined as:
\[\begin{equation} \( \dot{m} = \frac{m}{\Delta t} \) \[10pt] \[10pt]\end{equation}\]
where \(m\) is the mass and \(\Delta t\) is the time interval. In regards to ion thrusters, the mass flow rate describes the rate of fuel consumption, which reflects how much mass is lost as the xenon ions are expelled from the thruster over time. This rate is crucial for calculating not only the thrust but also for determining how long the spacecraft can operate before it runs out of fuel.

Thrust Calculation

Understanding Thrust

Thrust is the force that moves a spacecraft through space and is generated by the propulsion system, such as an ion thruster. We calculate thrust using the relationship:
\[\begin{equation} T = \dot{m} v \[10pt] \[10pt]\end{equation}\]
where \(T\) is the thrust produced, \(\dot{m}\) is the mass flow rate, and \(v\) is the exhaust velocity of the ions. This fundamental equation allows us to determine how efficiently a spacecraft can be propelled given a specific fuel consumption rate and ion ejection speed. By rearranging the equation to solve for \(\dot{m}\), as shown in the provided solution, engineers can determine the amount of fuel needed for a mission or the duration the spacecraft can operate with its fuel reserves.

Xenon Ion Propulsion

What is Xenon Ion Propulsion?

Xenon ion propulsion is a form of electric spacecraft propulsion system where xenon gas is ionized, then accelerated to high speeds using an electric field, effectively pushing the spacecraft in the opposite direction according to Newton's third law of motion. This advanced technology offers high efficiency and a better thrust-to-weight ratio compared to conventional chemical rockets.

Xenon as a Propellant

Xenon is often chosen as the propellant for ion thrusters due to its inert nature, high atomic mass, and the ease with which it can be ionized and accelerated. These factors make xenon an ideal candidate to produce a sufficient thrust while maintaining low fuel consumption, extending the life of space missions without the need for excessive amounts of propellant.