Question

An ion thruster mounted in a satellite uses electric forces to eject xenon ions with a speed of \(22.91 \mathrm{~km} / \mathrm{s}\). The ion thruster operates continuously while pointed in the same direction until it uses all \(25.95 \mathrm{~kg}\) of the available fuel. The change in speed of the satellite is \(275.0 \mathrm{~m} / \mathrm{s}\). What was the mass of the satellite and the fuel before the thruster started operating?

Short answer

Using the conservation of momentum principle, the initial mass of the satellite and fuel before the thruster started operating was approximately 21,653.9 kg.

Step-by-step solution

Identify given values

We are given the following values: - Xenon ion ejection speed from the thruster: \(v_1 = 22.91 \, \mathrm{km/s} = 22,910 \, \mathrm{m/s}\) - Total mass of fuel used: \(m_1 = 25.95 \, \mathrm{kg}\) - Change in speed of the satellite: \(\Delta v_2 = 275.0 \, \mathrm{m/s}\)

Apply conservation of momentum principle

According to the conservation of momentum principle, the total momentum before the ion thruster is turned on is equal to the total momentum after the thruster has been operating: Initial momentum = Final momentum \((m_1+m_2)\cdot 0 = m_1\cdot v_1 + m_2\cdot (v_2 - \Delta v_2)\) Here, \(m_2\) is the mass of the satellite, and \(v_2\) is its final velocity.

Solve for the mass of the satellite m_2

Rearrange the conservation of momentum equation to solve for \(m_2\): \(m_2 = \frac{m_1\cdot v_1}{\Delta v_2}\) Substitute the given values: \(m_2 = \frac{25.95 \, \mathrm{kg} \cdot 22,910 \, \mathrm{m/s}}{275.0 \, \mathrm{m/s}}\) \(m_2 \approx 21,627.95 \, \mathrm{kg}\)

Find the initial mass of the satellite and fuel

The initial mass of the satellite and the fuel is the sum of \(m_1\) and \(m_2\): Initial mass = \(m_1 + m_2\) Initial mass = \(25.95\, \mathrm{kg} + 21,627.95\, \mathrm{kg} \approx 21,653.9\, \mathrm{kg}\) Thus, the mass of the satellite and the fuel before the thruster started operating was approximately \(21,653.9\, \mathrm{kg}\).

Key concepts

Conservation of Momentum

The principle of conservation of momentum is critical to understanding how ion thrusters function in space. It states that, in an isolated system where there are no external forces, the total momentum before an interaction is equal to the total momentum after the interaction.

In the context of an ion thruster, which expels ions to propel a satellite, the momentum carried away by the ejected ions must be equal and opposite to the momentum gained by the satellite. This is why an ion thruster can change a satellite's velocity even in the vacuum of space, where there are no external references.

Satellite Propulsion

Satellite propulsion is a crucial component in orbital mechanics, and ion thrusters provide a very efficient method. They work by expelling ions (charged particles) at very high speeds to propel the satellite in the opposite direction, according to Newton's third law of motion: for every action, there is an equal and opposite reaction.

In a satellite equipped with an ion thruster, electricity is used to ionize a propellant—frequently xenon—creating positive ions. These ions are then accelerated by an electric field and ejected from the thruster at very high speeds, providing the propulsive force necessary for satellite maneuvering and position maintenance.

Xenon Ion Ejection

Xenon is commonly used in ion thrusters due to its inert nature and high atomic mass, which allows it to provide a greater momentum change per ion. The ejection process involves ionizing the xenon gas, then accelerating the ions through a grid system using an electric field.

The highly accelerated ions exit the thruster at speeds of tens of kilometers per second, and although the mass flow rate of the ions is low, the cumulative effect over time can significantly alter the satellite's velocity. This method of propulsion is known for its high specific impulse, which is a measure of propulsion efficiency, making it ideal for long-duration deep space missions.

Momentum Calculation

In a given exercise like ours, where an ion thruster is involved, momentum calculation allows us to determine quantities such as the mass of the satellite. Using the formula derived from the conservation of momentum, we can relate the mass and velocity of the expelled xenon ions to the subsequent change in velocity of the satellite.

By rearranging the momentum conservation equation, we can solve for the mass of the satellite before fuel ejection. In our case, we deduced the mass by dividing the product of the mass of ejected fuel and its ejection velocity by the change in speed of the satellite. Mathematically, this becomes:\[m_2 = \frac{m_1 \cdot v_1}{\Delta v_2}\]. The calculation demonstrates that even when dealing with high-speed particles and large masses, the concepts of physics provide a coherent path to finding solutions.