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Chapter 6: Problem 49

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?

Short Answer

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

Step by step solution

01

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.
02

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.
03

Compute Gravitational Acceleration

Calculate the gravitational acceleration \( g \) on Europa: \[ g = \frac{6.674 \times 10^{-11} \cdot 4.78 \times 10^{22}}{(1.565 \times 10^6)^2} \approx 1.314 \, \mathrm{m/s^2} \]
04

Calculate Maximum Walking Speed

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥檚 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/s虏 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.

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Most popular questions from this chapter

Your spaceship lands on an unknown planet. To determine the characteristics of this planet, you drop a wrench from \(5.00 \mathrm{~m}\) above the ground and measure that it hits the ground 0.811 s later. (a) What is the acceleration of gravity near the surface of this planet? (b) Assuming that the planet has the same density as that of earth \(\left(5500 \mathrm{~kg} / \mathrm{m}^{3}\right),\) what is the radius of the planet?

A \(52 \mathrm{~kg}\) ice skater spins about a vertical axis through her body with her arms horizontally outstretched, making 2.0 turns each second. The distance from one hand to the other is \(1.50 \mathrm{~m}\). Biometric measurements indicate that each hand typically makes up about \(1.25 \%\) of body weight. (a) Draw a free-body diagram of one of her hands. (b) What horizontal force must her wrist exert on her hand? (c) Express the force in part (b) as a multiple of the weight of her hand.

The star Rho' Cancri is 57 light-years from the earth and has a mass 0.85 times that of our sun. A planet has been detected in a circular orbit around Rho \({ }^{1}\) Cancri with an orbital radius equal to 0.11 times the radius of the earth's orbit around the sun. What are (a) the orbital speed and (b) the orbital period of the planet of Rho \(^{1}\) Cancri?

Observations of this planet over time show that it is in a nearly circular orbit around its star and completes one orbit in only 9.5 days. How many times the orbital radius \(r\) of the earth around our sun is this exoplanet's orbital radius around its sun? Assume that the earth is also in a nearly circular orbit. A. \(0.026 r\) B. \(0.078 r\) C. \(0.70 r\) D. \(2.3 r\)

Neutron stars, such as the one at the center of the Crab Nebula, have about the same mass as our sun, but a much smaller diameter. If you weigh \(675 \mathrm{~N}\) on the earth, what would you weigh on the surface of a neutron star that has the same mass as our sun and a diameter of \(20.0 \mathrm{~km} ?\)
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