Problem 30 - College Physics

Question

Space station. You are designing a space station and want to get some idea how large it should be to provide adequate air for the astronauts. Normally, the air is replenished, but for security, you decide that there should be enough to last for two weeks in case of a malfunction. (a) Estimate how many cubic meters of air an average person breathes in two weeks. A typical human breathes about 1\(/ 2 \mathrm{L}\) per breath. (b) If the space station is to be spherical, what should be its diameter to contain all the air you calculated in part (a)?

Step-by-Step Solution

Answer

The space station diameter should be approximately 6.74 meters.

Step 1: Understand the Problem

We need to estimate the total volume of air a person requires to breathe over two weeks and then calculate the size of a spherical space station to hold that volume.

Step 2: Calculate Breathing Volume per Person

First, determine how much air a person breathes per minute. Since a person breathes 0.5 liters per breath and assumes 12 breaths per minute, we multiply: \(0.5 \text{ L/breath} \times 12 \text{ breaths/minute} = 6 \text{ L/minute}\).

Step 3: Convert to Cubic Meters

Next, convert this breathing rate to cubic meters: \(6 \text{ L/min} = 0.006 \text{ m}^3\text{/min}\) (since 1 cubic meter = 1000 liters).

Step 4: Calculate Total Breathing Volume for Two Weeks

Determine how many minutes there are in two weeks: \(14 \text{ days} \times 24 \text{ hours/day} \times 60 \text{ minutes/hour} = 20160 \text{ minutes}\). Multiply by the breathing rate: \(20160 \text{ minutes} \times 0.006 \text{ m}^3/ ext{minute} = 120.96 \text{ m}^3\). So each person requires approximately \(120.96 \text{ m}^3\) of air for two weeks.

Step 5: Calculate Volume of Spherical Space Station

Use the formula for the volume of a sphere: \( V = \frac{4}{3} \pi r^3 \). Set \( V = 120.96 \) to solve for radius \( r \).

Step 6: Solve for Radius

Rearrange the formula to \( r^3 = \frac{3V}{4\pi} = \frac{3 \times 120.96}{4\pi} \). Calculate \( \approx 38.5 \). Thus, \( r = (38.5)^{1/3} \approx 3.37 \text{ m} \).

Step 7: Calculate Diameter

The diameter of the sphere is twice the radius. Therefore, \( \text{Diameter} = 2 \times 3.37 \approx 6.74 \text{ m} \).

Key Concepts

Volume Estimation

Volume estimation is crucial when planning structures like a space station. It involves calculating the amount of space needed for different components, in this case, for air that supports human life. To start with, consider the average breathing rate: a human breathes about 0.5 liters of air per breath.

Given that on average, a person takes 12 breaths per minute, you find the breathing volume as follows:

• Breaths per minute: 12
• Liters per breath: 0.5
• Calculate: 0.5 L/breath × 12 breaths/minute = 6 L/min

Next, you need the total volume of air needed for two weeks. Use the formula:

• Minutes in two weeks: 14 days × 24 hours/day × 60 minutes/hour = 20,160 minutes
• Total volume: 6 L/min × 20,160 minutes = 120,960 liters = 120.96 cubic meters

This volume ensures that astronauts have enough breathing air.

Spherical Geometry

Spherical geometry refers to the mathematics of spheres. This is crucial for designing a spherical space station. The formula for the volume of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]
Where \( V \) is the volume and \( r \) is the radius. Let's say the station needs to store the calculated 120.96 cubic meters of air:

• Set \( V = 120.96 \) m³
• Rearrange to find \( r \): \( r^3 = \frac{3V}{4\pi} \)
• Calculate: \( r^3 = \frac{3 \times 120.96}{4\pi} \approx 38.5 \)
• Find radius: \( r = (38.5)^{1/3} \approx 3.37 \) meters

Spheres efficiently enclose volume, which is why they're a preferred shape in space design.

Breathing Rate Calculations

Breathing rate calculations help determine how much air an astronaut needs. This is pivotal when estimating the size of life-support systems.
To calculate this, start with the breathing rate:

• Average liters per breath: 0.5 L
• Breaths per minute: 12
• Breathing volume: 0.5 L × 12 breaths/min = 6 L/min

Over two weeks:

• Minutes in two weeks: 20,160
• Total volume: 6 L/min × 20,160 minutes = 120,960 liters or 120.96 m³

Getting an accurate measure of these needs is crucial to safety.

Unit Conversion

Unit conversion allows us to switch between different measurement units. This is essential in space station design, where accurate dimensions are needed.
Consider air volume, initially calculated in liters:

• 1 cubic meter (m³) = 1,000 liters (L)
• Breathing rate: 6 L/min
• Converted rate: 6 L/min = 0.006 m³/min

Unit conversion is fundamental in ensuring that every calculation aligns with real-world measurements when designing a space station.

Safety Measures in Space Stations

Safety measures are central in space station design, especially when concerning oxygen supply.

• Ensure there's a fail-safe for air replenishment.
• Calculate a surplus air volume, supporting astronauts if systems fail.
• Design backup systems that operate independently from primary ones.

Planning safety measures ensures astronauts' well-being in self-contained environments.