Question

An astronaut is performing a space walk outside the International Space Station. The total mass of the astronaut with her space suit and all her gear is 115 kg. A small leak develops in her propulsion system, and 7.00 g of gas is ejected each second into space with a speed of \(800 . \mathrm{m} / \mathrm{s}\). She notices the leak 6.00 s after it starts. How much will the gas leak have caused her to move from her original location in space by that time?

Short answer

Answer: The astronaut will have moved approximately 0.8772 meters away from her original location.

Step-by-step solution

Determine net force due to the ejected gas

In order to determine the net force experienced by the astronaut, we need to calculate the momentum change of the gases being ejected per second. Momentum can be calculated using the formula: Momentum = Mass × Velocity The mass of the gas ejected per second is \(7.00~g = 0.007~kg\), and the velocity is \(800~m/s\).

Calculate the net force on the astronaut

Using the momentum equation, we can find the net force: Net Force = (Mass of gas/sec) × (Velocity of gas) Net Force = \(0.007~kg × 800~m/s = 5.6~N\) The net force of \(5.6~N\) is acting opposite to the direction of the ejected gas.

Calculate the astronaut's acceleration

Now, we can calculate the astronaut's acceleration due to the net force. We use Newton's second law of motion: Newtons's law states: \(F=ma\) Where \(F\) is the net force, \(m\) is the total mass of the astronaut and \(a\) is the acceleration of the astronaut. Solving for acceleration \(a\), we get: \(a = \frac{F}{m}\) Using the values provided (Net Force = \(5.6~N\) and Mass of astronaut = \(115~kg\)), we can calculate the value of \(a\): \(a = \frac{5.6~N}{115~kg} = 0.0487~m/s^2\)

How much the gas leak caused the astronaut to move

With the astronaut's acceleration determined, we can now find out their displacement (how much they have moved) during the 6 seconds from when the gas leak started. For this, we'll use the equation: Displacement = Initial velocity × Time + \(\frac{1}{2}\) × Acceleration × (Time)\(^2\) The astronaut is initially at rest, so the initial velocity is zero. Therefore, the equation simplifies to: Displacement = \(\frac{1}{2}\) × \(0.0487~m/s^2\) × (\(6~s)^2\) Displacement = \(0.5 × 0.0487 × 36 = 0.8772~m\) The astronaut will have moved approximately \(0.8772~m\) away from her original location in space after 6 seconds since the gas leak started.

Key concepts

Momentum Conservation

When considering how an astronaut moves in space due to a gas leak, we must invoke the law of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant if no external forces are acting on it.

Imagine the astronaut and her gear as one system and the gas as another. Before the leak, both the astronaut and the gas are at rest relative to one another, meaning the system's total momentum is zero. When gas leaks and shoots out one way, the astronaut must move in the opposite direction to keep the total momentum of the system at zero. This is a fundamental principle, which explains why the astronaut begins to drift slowly in the opposite direction of the ejected gas.

The key equation here is: \( Momentum = Mass \times Velocity \). The momentum of the gas is the product of its mass and velocity as it leaves the propulsion system. By conservation of momentum, this must be equal and opposite to the momentum gained by the astronaut.

Newton's Second Law of Motion

Newton's second law provides the relationship between an object's motion and the forces acting on it. It's best summarized by the equation: \(F = ma\), where \(F\) stands for force, \(m\) for mass, and \(a\) for acceleration.

In the case of our spacewalking astronaut, the force experienced by her due to the gas ejection is causing a change in her velocity, or in other words, she is accelerating. The force calculated from the momentum of the gas is used to determine the astronaut's acceleration by reworking the second law to \(a = \frac{F}{m}\). Even in the absence of gravity, an astronaut in space can accelerate due to external forces—such as those from leaking gas—demonstrating Newton's law at work.

Kinematic Equations

Kinematics describes motion, and the equations of kinematics relate the variables of motion to one another. Each variable represents a specific aspect of motion—displacement (\(s\)), initial velocity (\(u\)), final velocity (\(v\)), acceleration (\(a\)), and time (\(t\)). One of these equations allows us to calculate the displacement of an object when we know its initial velocity, acceleration, and the time taken: \( s = ut + \frac{1}{2} at^2 \).

Our astronaut started at rest, making her initial velocity zero. Thus, the displacement due only to the acceleration (caused by the leaking gas) over the time interval of 6 seconds is what we calculate. This kinematic equation is particularly useful for understanding motion in space where there are no roads or landmarks to gauge movement. It offers a clear mathematical framework to describe the impact of forces (like the leaking gas) on the astronaut's position.

Astronaut Propulsion

Astronaut propulsion in space is a practical application of Newton's third law of motion: for every action, there is an equal and opposite reaction. When our astronaut experiences the gas leak, the action of gas being expelled backward results in a reaction that propels the astronaut forward.

In the vacuum of space, where there is no air resistance, even a small amount of force can accelerate a spacewalker. The propulsion system on a spacesuit typically uses bursts of gas to control and maneuver the astronaut. However, even a single consistent force, like that from the leak, can move the astronaut a significant distance.

This concept reinforces the importance of understanding how actions in one direction will lead to movements in the opposite one, which is crucial for precise movement and control during a spacewalk.