Question

An ion thruster mounted in a satellite with mass \(2149 \mathrm{~kg}\) (including \(23.37 \mathrm{~kg}\) of fuel) uses electric forces to eject xenon ions with a speed of \(28.33 \mathrm{~km} / \mathrm{s}\). If the ion thruster operates continuously while pointed in the same direction until it uses all of the fuel, what is the change in the speed of the satellite?

Short answer

Answer: Approximately 307.74 m/s.

Step-by-step solution

Identify the initial and final conditions

Initially, the satellite has a certain mass which includes the mass of the fuel. After operating the ion thruster continuously until all of the fuel is consumed, the satellite's mass will be reduced (by the mass of the fuel), and its speed will change.

Apply the conservation of momentum principle

The conservation of momentum principle states that the total momentum of a closed system remains constant if no external forces act on it. In this case, the closed system is the satellite and ejected fuel. Let's denote the initial momentum as \(p_i\) and final momentum as \(p_f\). Then we can write the momentum conservation equation as: \(p_i = p_f\)

Separate the initial and final momenta

The initial momentum of the system is comprised of the satellite's mass, \(m_{satellite}\), and its initial speed \(v_i\), which we'll assume to be zero: \(p_i = m_{satellite} \cdot v_i\) The final momentum consists of the momentum of both the satellite minus the consumed fuel and the momentum of the ejected fuel mass. We'll denote the mass of the fuel as \(m_{fuel}\), the final speed of the satellite as \(v_f\), and the speed of the ejected xenon ions as \(v_{ions}\). The final momentum can then be written as: \(p_f = (m_{satellite} - m_{fuel}) \cdot v_f + m_{fuel} \cdot v_{ions}\)

Substitute the known values and solve for \(v_f\)

Now we can substitute the known values into the conservation of momentum equation and solve for the change in speed of the satellite, \(v_f\): \(2149 \, kg \cdot 0 = (2149 \, kg - 23.37 \, kg) \cdot v_f + 23.37 \, kg \cdot (28.33 \cdot 10^3 \, m/s)\) We have: \(0 = 2125.63 \, kg \cdot v_f + 23.37 \, kg \cdot (28.33 \cdot 10^3\, m/s)\) Now solve for \(v_f\): \(v_f = \frac{23.37 \, kg \cdot (28.33 \cdot 10^3\, m/s)}{2125.63 \, kg}\) \(v_f \approx \boxed{307.74\, m/s}\) So, the change in the speed of the satellite after all the fuel is used is approximately \(307.74\, m/s\).

Key concepts

Ion Thruster Physics

Ion thrusters represent a sophisticated technology used to propel spacecraft by ejecting ions at high velocities. But how do they work? An ion thruster generates thrust by accelerating ions using electric fields. These ions are produced in the thruster by ionizing a propellant gas, usually xenon, which is inert and has a high atomic mass. The ionization process creates positively charged ions and electrons. The ions are then accelerated by the electric field to high speeds and ejected out of the thruster, creating thrust.

The acceleration of ions in an ion thruster is a classic example of Newton's third law of motion, which states that for every action, there's an equal and opposite reaction. As ions are expelled at high speed in one direction, the spacecraft is pushed in the opposite direction. This mechanism allows for very efficient space travel, especially for long-duration missions. Important factors in ion thruster physics include the efficiency of ionization, the electric potential difference used for acceleration, and the mass of the ions.

Momentum Conservation in Space

Understanding momentum conservation in space is crucial for predicting the movement of spacecraft and planning missions. As per the principle of momentum conservation, in the absence of external forces, the total momentum of a system remains unchanged. In the vacuum of space, there are negligible external forces once a spacecraft is in motion, making this principle particularly relevant.

In the problem given, the absent external forces allow us to use this principle to determine the satellite's new velocity after the ion thruster operation. The exercise exemplifies that as ions are ejected rearward, the satellite must move forward to conserve the system's overall momentum. This behavior aligns with the idea that the momentum before and after the thruster operation must be equal, leading to our momentum conservation equation. This principle extends beyond ion thrusters and is fundamental in understanding any space propulsion system or collision.

Satellite Mass and Velocity

The relation between a satellite's mass and velocity is an integral part of understanding its dynamics in orbit. Initially, the satellite is at rest with a total mass that includes the fuel. As the fuel is ejected, the satellite's mass decreases, but due to momentum conservation, its speed increases.

To calculate the change in the satellite's speed, we consider the velocity of the ejected ions and the mass of fuel consumed, as detailed in steps 1 through 4 of the provided solution. With the ion thruster ejecting ions at a specific speed, in this case, 28.33 km/s, and knowing the total mass of fuel, we can determine the satellite's change in speed by applying the principle of momentum conservation. Therefore, the satellite's final velocity is intrinsically linked to both the mass of the consumed fuel and the velocity of the ejected propellant.