Problem 44 - College Physics

Question

Jupiter's moon Io has active volcanoes (in fact, it is the most volcanically active body in the solar system) that eject material as high as \(500 \mathrm{~km}\) (or even higher) above the surface. Io has a mass of \(8.94 \times 10^{22} \mathrm{~kg}\) and a radius of \(1815 \mathrm{~km}\). Ignore any variation in gravity over the \(500 \mathrm{~km}\) range of the debris. How high would this material go on earth if it were ejected with the same speed as on lo?

Step-by-Step Solution

Answer

The material would rise approximately 125 km on Earth if ejected with the same speed as on Io.

Step 1: Determine the gravity on Io

Compute the gravitational acceleration on Io using the formula \( g = \frac{G \, M}{R^2} \), where \(G\) is the gravitational constant \(6.674 \times 10^{-11} \, \text{m}^3\, \text{kg}^{-1}\, \text{s}^{-2}\), \(M\) is the mass of Io \(8.94 \times 10^{22} \, \text{kg}\), and \(R\) is the radius of Io \(1815 \times 10^3 \, \text{m}\). Calculate \(g\) for Io.

Step 2: Calculate the escape speed from Io

Using the gravitational acceleration from Step 1, calculate the initial speed (\(v_0\)) required to reach a height of 500 km on Io using the equation for free fall \( h = \frac{v_0^2}{2g} \). Rearrange to solve for \(v_0\) when \(h = 500 \times 10^3 \, \text{m}\).

Step 3: Determine the same initial speed's effect on Earth

With the calculated speed \(v_0\) from Step 2, determine how high the debris would travel on Earth. Use Earth’s gravitational acceleration \(g_e = 9.81 \, \text{m/s}^2\) in the free fall equation \( h = \frac{v_0^2}{2g_e} \) to calculate the height \(h\) on Earth.

Key Concepts

Jupiter's moon Io

Jupiter's moon Io is a fascinating celestial body orbiting the gas giant Jupiter. Among its many intriguing characteristics, Io is renowned for its extreme geological activity. It is the most volcanically active moon in our solar system, even more active than Earth. This intense volcanic activity is primarily due to the gravitational interaction between Jupiter, Io, and other Galilean moons, which causes significant tidal heating within Io's interior, thereby driving its volcanic phenomena. Natural satellites like Io are of keen interest to scientists. They help us understand volcanic processes and also provide insights into the interactions within planetary systems.
Io’s volcanic activity significantly affects its landscape, creating a dynamically changing surface. This moon is not only a source of awe but also a subject of numerous studies to understand the mechanisms driving such unique and intense volcanic activities.

Volcanic activity

Io's volcanic activity sets it apart from any other celestial body in our solar system. The moon experiences substantial amounts of volcanic eruptions, which spew lava and other materials high up into space. These eruptions can reach heights of up to 500 kilometers or more. Such activity is fueled by molten silicate magma chambers below Io's surface, heated by tidal forces. This makes Io an excellent natural laboratory for studying volcanic processes.
Volcanic activity on Io results in cloud-like plumes that rise high into the atmosphere and then snow down as sulfur-dioxide frost. This continuous activity reshapes the surface of Io, making it one of the most geologically active places discovered. By studying Io, researchers gain insights into tectonic activities and thermal evolution of rocky bodies.

Gravitational acceleration

Gravitational acceleration is a crucial concept in understanding how celestial bodies like Io function. It refers to the acceleration due to gravity at the surface of a celestial body. For Io, we calculate this value to understand how the gravitational forces affect the volcanoes' ejected materials. The formula for gravitational acceleration is given by \[ g = \frac{G \cdot M}{R^2} \]where

• \(G\) is the gravitational constant, approximately \(6.674 \times 10^{-11} \, \text{m}^3\, \text{kg}^{-1}\, \text{s}^{-2}\)
• \(M\) is the mass of Io, estimated at \(8.94 \times 10^{22} \, \text{kg}\)
• \(R\) is the radius of Io, about \(1815 \times 10^3 \, \text{m}\).

Thus, gravitational acceleration helps estimate the speed needed for materials to reach considerable heights and to predict how these processes might differ on other planets with different gravitational forces.

Free fall motion

Free fall motion is an important concept to grasp when studying the physics of volcanic eruptions on celestial bodies like Io. In this context, free fall motion refers to the movement of volcanic debris ejected from the moon’s surface, which rises and eventually falls back due to gravitational pull. The equation for free fall, given by \[ h = \frac{v_0^2}{2g} \]where

• \(h\) is the height reached,
• \(v_0\) is the initial velocity of the ejected material, and
• \(g\) is the gravitational acceleration.

This equation allows astronomers to calculate how high volcanic material can reach based on initial speed and gravitational conditions on Io. Furthermore, when understanding planetary differences, this formula helps compare outcomes on different planets, like calculating an equivalent height if the same material were ejected with similar speed on Earth.