Question

An ion thruster mounted in a satellite with mass \(2161 \mathrm{~kg}\) (including fuel) uses electric forces to eject xenon ions with a speed of \(20.61 \mathrm{~km} / \mathrm{s}\) The ion thruster operates continuously while pointed in the same direction until it uses all the available fuel. The change in the speed of the satellite is \(236.4 \mathrm{~m} / \mathrm{s} .\) How much fuel was available to the thruster?

Short answer

Answer: Approximately 24.8 kg.

Step-by-step solution

Understand the problem

We are asked to find the mass of fuel available to the thruster of a satellite. To do this, we can use the given change in speed and the principles of conservation of momentum.

Write down the conservation of momentum principle

Since the momentum of the satellite-ion system is conserved, we can write the equation for conservation of momentum as: Mass_s * ΔV_s = Mass_f * V_f Where: Mass_s = mass of the satellite ΔV_s = change in the speed of the satellite Mass_f = mass of the ejected fuel V_f = ejection speed of the fuel

Substitute given values in the equation

Let's substitute the given values in the conservation of momentum equation: \(2161 \mathrm{~kg}\) * \(236.4 \mathrm{~m} / \mathrm{s}\) = Mass_f * \(20610 \mathrm{~m} / \mathrm{s}\)

Solve for Mass_f (mass of ejected fuel)

Now let's solve for Mass_f to find the mass of the ejected fuel: \(\frac{2161 \mathrm{~kg} * 236.4 \mathrm{~m} / \mathrm{s}}{20610 \mathrm{~m} / \mathrm{s}} = \text{Mass_f}\) Mass_f = \(\frac{510847.4}{20610} \approx 24.8 \mathrm{~kg}\) So, the mass of the available fuel to the ion thruster was approximately 24.8 kg.

Key concepts

Physics Problem Solving

The beauty of physics problem solving lies within its systematic approach to unravel mysteries of the natural world. In the context of our exercise, the problem deals with a practical application of physics — an ion thruster on a satellite. To tackle such a problem, one needs to have a clear understanding of both the physical principles involved and their mathematical description.

When endeavoring to solve physics problems, a step-by-step process is crucial. As showcased in our exercise, the first step is to understand the problem and identify the core principle that applies - in this case, the conservation of momentum. Next, we move onto writing down the related physics equation, and subsequently, substituting the known values into the equation. Finally, we solve for the unknown variable.

This methodical approach not only simplifies complex problems but also teaches us the importance of applying theoretical knowledge to real-world scenarios, reinforcing the concept that physics is deeply embedded in our daily lives.

Applying the Method

Specifically, in our example, we leverage the conservation of momentum to determine an unknown quantity — the mass of the fuel used by the thruster. Such skillful problem solving encourages critical thinking and a deeper understanding of the laws that govern the movement and interaction of objects.

Ion Thruster Physics

Ion thrusters represent a cutting-edge technology primarily used in space exploration. They operate on the principle of ionizing a propellant, commonly xenon, and then using electric fields to accelerate these ions to extremely high velocities. The thrust generated by these ionized particles being ejected is the key mechanism for propulsion in space, where traditional combustion engines are ineffectual.

The ion thruster's efficiency comes from the high speed at which it can expel ions, which, as our exercise mentions, can reach up to 20.61 km/s. In the vacuum of space, the engine doesn't have to overcome air resistance, allowing every ejected ion to contribute to the satellite's change in momentum.

Understanding the physics behind ion thrusters not only gives insight into how we navigate the vastness of space but also demonstrates the application of core physics principles, like conservation of momentum, in advanced technological settings.

The Thrust Equation

The thrust of an ion thruster can be equated to the rate of momentum change of the ions as they are expelled from the thruster, which our exercise encapsulates through the momentum conservation equation.

Momentum Conservation in Space

Momentum conservation is a fundamental concept in physics, implying that in a closed system with no external forces, the total momentum remains constant over time. In the vacuum of space, where friction and air resistance are absent, this principle is particularly evident.

The satellite with an ion thruster, as presented in our problem, is an excellent demonstration of this principle. When ions are ejected from the thruster, they carry away momentum, and hence, by the law of conservation of momentum, the satellite must gain momentum in the opposite direction. The lack of external forces in space makes it an ideal environment to observe pure momentum conservation.

By harnessing this principle, spacecraft can maneuver and accelerate in space with precision, enabling long-term missions and profound exploration of our universe. Understanding momentum conservation in such contexts helps us appreciate the natural laws that make space travel possible.

Cost-Effective Acceleration

Moreover, this principle allows for efficient use of fuel in space, as every action has an equal and opposite reaction, hence optimizing the mass and velocity of ejected ions for maximal thrust.