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Chapter 3: Problem 4

An astronaut working with many tools some distance away from a spacecraft is stranded when the "maneuvering unit" malfunctions. How can the astronaut return to the spacecraft by sacrificing some of the tools?

Short Answer

Expert verified
Answer: To find the necessary velocity to propel themselves back towards the spacecraft (v2_astronaut), the astronaut can apply the conservation of momentum principle and use the equation: v2_astronaut = (m_tools * v2_tools) / (m - m_tools), where m is the total mass of the astronaut and their tools, m_tools is the mass of the discarded tools, and v2_tools is the velocity at which the tools are thrown. By calculating v2_astronaut, the astronaut can determine how fast and in which direction to throw the tools to create enough momentum to return to the spacecraft.

Step by step solution

01

Identify the principle of conservation of momentum

The principle of conservation of momentum states that the total momentum of a closed system remains constant if no external forces are acting on it. This will be the guiding concept throughout the process of solving this problem.
02

Calculate the initial momentum of the astronaut and tools

Let's denote the total mass of the astronaut and their tools as "m" and their initial velocity as "v1" (which is 0 in this case, since they are stationary). According to the conservation of momentum, the initial momentum of the astronaut and their tools can be written as: Initial momentum = m * v1 In this case, Initial momentum = 0
03

Calculate the final momentum of the astronaut and tools after sacrificing some tools

In order to return to the spacecraft, the astronaut will have to throw some tools away from the spacecraft. Let's denote the mass of the discarded tools as "m_tools" and the velocity at which they are thrown as "v2_tools". The final momentum of the discarded tools can be expressed as: Final momentum of tools = m_tools * v2_tools Now, let's denote the final velocity of the astronaut (without the discarded tools) as "v2_astronaut". The final momentum of the astronaut can be calculated as follows: Final momentum of astronaut = (m - m_tools) * v2_astronaut
04

Apply conservation of momentum

According to the conservation of momentum, the initial momentum of the system will be equal to the final momentum. Since at the beginning the astronaut is not moving, initial momentum is 0. 0 = (m - m_tools) * v2_astronaut - m_tools * v2_tools
05

Find the desired velocity for the astronaut

To find the desired velocity for the astronaut (v2_astronaut), we need to rearrange the equation from Step 4. v2_astronaut = (m_tools * v2_tools) / (m - m_tools) By calculating the value of v2_astronaut, the astronaut can find out how fast they need to throw the tools and in which direction, in order to generate enough momentum to propel themselves back towards the spacecraft.

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