Problem 16 - University Physics with Modern 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 km (or even higher) above the surface. Io has a mass of 8.93 \(\times\) 10\(^{22}\) kg and a radius of 1821 km. For this calculation, ignore any variation in gravity over the 500-km range of the debris. How high would this material go on earth if it were ejected with the same speed as on Io?

Step-by-Step Solution

Answer

The material would reach approximately 91.6 km on Earth.

Step 1: Calculate Escape Velocity on Io

The escape velocity on a celestial body can be calculated using the formula \( v = \sqrt{2g_i h} \), where \( g_i \) is the acceleration due to gravity on Io, and \( h \) is the maximum height reached by the material, which is 500 km. First, we need to calculate \( g_i \) using \( g_i = \frac{G M_i}{R_i^2} \). Given \( G = 6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2 \), \( M_i = 8.93 \times 10^{22} \text{ kg} \), and \( R_i = 1821 \text{ km} = 1.821 \times 10^6 \text{ m} \), we have:\[ g_i = \frac{6.674 \times 10^{-11} \cdot 8.93 \times 10^{22}}{(1.821 \times 10^6)^2} = 1.796 \text{ m/s}^2 \]

Step 2: Calculate Initial Speed on Io

To find the initial speed \( v_i \) on Io, use the formula \( v_i = \sqrt{2g_i h} \). Substituting, we get:\[ v_i = \sqrt{2 \cdot 1.796 \cdot 500,000} = \sqrt{1,796,000} \approx 1341 \text{ m/s} \]

Step 3: Calculate Maximum Height on Earth

On Earth, the same initial speed \( v_i \) would propel the debris to a different height due to Earth's stronger gravity. Use the formula \( h_e = \frac{v_i^2}{2g_e} \), where \( g_e = 9.81 \text{ m/s}^2 \). Calculate \( h_e \):\[ h_e = \frac{1341^2}{2 \cdot 9.81} = \frac{1797481}{19.62} \approx 91590 \text{ m} = 91.6 \text{ km} \]

Key Concepts

Gravity on Io

Io, one of Jupiter's moons, is known for its intense volcanic activity. This activity can cause materials to be ejected high above its surface. Understanding gravity on Io is crucial for calculating the maximum height materials can reach. To determine the gravity on Io (\( g_i \),), you can employ the gravitational formula:

• \( g_i = \frac{GM_i}{R_i^2} \)

where:

• \( G \) is the gravitational constant \((6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2) \)
• \( M_i \) is Io's mass \((8.93 \times 10^{22} \, \text{kg}) \)
• \( R_i \) is Io's radius \((1.821 \times 10^6 \, \text{m}) \)

Plugging these values in, you find that the gravitational acceleration on Io is approximately \( 1.796 \, \text{m/s}^2 \). This value is quite low compared to Earth's gravity, emphasizing why the material can reach significant altitudes on Io.

Initial Speed Calculation

To understand how far materials ejected from Io can travel, it's imperative to calculate the initial speed. This is done using the formula:

• \( v_i = \sqrt{2g_ih} \)

where:

• \( g_i \) is Io's gravity \( (1.796 \, \text{m/s}^2) \)
• \( h \) is the maximum ejected height \((500 \, \text{km} = 500,000 \, \text{m}) \)

By substituting:\[v_i = \sqrt{2 \cdot 1.796 \cdot 500,000} \approx 1341 \, \text{m/s}\]This value represents the initial speed of the material as it leaves the volcanic surface of Io. Understanding this speed helps compare how material behaves when ejected into different gravitational environments.

Maximum Height Calculation on Earth

Once the initial speed is known, it's insightful to predict how high the debris would reach on Earth due to differences in gravitational strength. Earth's stronger gravity \(( g_e = 9.81 \, \text{m/s}^2)\) affects the maximum height the same speed can achieve. The formula used is:

• \( h_e = \frac{v_i^2}{2g_e} \)

Substituting the known values:\[h_e = \frac{1341^2}{2 \cdot 9.81} \approx 91.6 \, \text{km}\]This calculation shows that under Earth's gravity, the volcanic material would reach a much lower peak compared to Io. This is due to Earth's gravity being significantly stronger, thereby pulling objects down more forcefully. Understanding these differences in maximum heights provides insight into how gravitational strength influences the motion of ejected materials.