Problem 49 - College Physics

Question

Europa, a satellite of Jupiter, is believed to have an ocean of liquid water (with the possibility of life) beneath its icy surface. (See Figure \(6.31 .)\) Europa is \(3130 \mathrm{~km}\) in diameter and has a mass of \(4.78 \times 10^{22} \mathrm{~kg} .\) In the future, we will surely want to send astronauts to investigate Europa. In planning such a future mission, what is the fastest that such an astronaut could walk on the surface of Europa if her legs are \(1.0 \mathrm{~m}\) long?

Step-by-Step Solution

Answer

The maximum walking speed on Europa is approximately 1.146 m/s.

Step 1: Understand Critical Velocity Formula

To find the maximum walking speed, we use the formula for the maximum speed of a pendulum or a simple swing: \[ v = \sqrt{g \cdot l} \] where \( v \) is the maximum speed, \( g \) is the gravitational acceleration on Europa, and \( l \) is the leg length of the astronaut.

Step 2: Calculate Gravitational Force on Europa

First, we find the gravitational acceleration \( g \) on Europa using the formula: \[ g = \frac{G \cdot M}{R^2} \] where \( G = 6.674 \times 10^{-11} \, \mathrm{m}^3/\mathrm{kg} \cdot \mathrm{s}^2 \) is the gravitational constant, \( M = 4.78 \times 10^{22} \mathrm{~kg} \) is the mass of Europa, and \( R = 3130/2 = 1565 \mathrm{~km} = 1.565 \times 10^6 \mathrm{~m} \) is the radius of Europa.

Step 3: Compute Gravitational Acceleration

Calculate the gravitational acceleration \( g \) on Europa: \[ g = \frac{6.674 \times 10^{-11} \cdot 4.78 \times 10^{22}}{(1.565 \times 10^6)^2} \approx 1.314 \, \mathrm{m/s^2} \]

Step 4: Calculate Maximum Walking Speed

With \( g \) and \( l = 1.0 \, \mathrm{m} \), compute the maximum speed: \[ v = \sqrt{1.314 \cdot 1.0} = \sqrt{1.314} \approx 1.146 \, \mathrm{m/s} \]

Key Concepts

Gravitational acceleration

Gravitational acceleration is the rate at which an object accelerates due to the gravitational pull of a celestial body. This concept is crucial when considering movement on other planets or moons, like Europa, Jupiter's fascinating moon. Europa's gravitational pull is less than that of Earth due to its smaller mass and size. To calculate gravitational acceleration on Europa, use the formula:

• \( g = \frac{G \cdot M}{R^2} \)

where:

• \( g \) represents gravitational acceleration,
• \( G \) is the universal gravitational constant \( 6.674 \times 10^{-11} \, \mathrm{m^3/kg \cdot s^2} \),
• \( M \) is Europa’s mass \( 4.78 \times 10^{22} \, \mathrm{kg} \),
• \( R \) is the radius of Europa \( 1.565 \times 10^6 \, \mathrm{m} \).

Calculating this, we find that the gravitational pull is approximately \( 1.314 \, \mathrm{m/s^2} \), much lower than Earth's \( 9.81 \, \mathrm{m/s^2} \). This affects how astronauts would experience walking on its surface.

Maximum walking speed

The maximum walking speed on a celestial body like Europa is determined by the gravitational acceleration and the astronaut's leg length. The physics behind it is similar to the pendulum’s motion. To find this speed, you use the formula:

• \( v = \sqrt{g \cdot l} \)

where:

• \( v \) is the maximum walking speed,
• \( g \) is the gravitational acceleration (1.314 m/s² on Europa),
• \( l \) is the length of the astronaut's legs (1.0 m in this scenario).

When we calculate this, \( v = \sqrt{1.314 \cdot 1.0} \approx 1.146 \, \mathrm{m/s} \). This relatively slower speed compared to Earth is due to the lesser gravitational force, meaning an astronaut could safely walk at this speed without bouncing off the icy surface.

Europa

Europa, one of Jupiter's largest moons, is an intriguing destination for future space exploration due to the potential presence of a subsurface ocean beneath its icy crust. Understanding its physical characteristics is essential for planning any missions.

• Diameter: 3130 km, making it one of the smaller Galilean moons.
• Mass: \( 4.78 \times 10^{22} \, \mathrm{kg} \), it has significantly less mass than Earth.
• Surface: Icy and smooth, with very few craters compared to other moons, hinting at a relatively young surface.

These characteristics influence both the gravitational forces and conditions for movement on its surface. Planning to explore Europa involves calculating gravitational effects to ensure astronaut safety and mobility, particularly due to its unique surface conditions and lower gravity.