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 × 10²² 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?

Verified step by step guidance

1. First, calculate the gravitational acceleration on Io using the formula: \( g_{Io} = \frac{G \cdot M_{Io}}{R_{Io}^2} \), where \( G \) is the gravitational constant \( 6.674 \times 10^{-11} \text{Nm}^2/\text{kg}^2 \), \( M_{Io} = 8.93 \times 10^{22} \text{kg} \), and \( R_{Io} = 1821 \times 10^3 \text{m} \).
2. Next, determine the initial velocity \( v_0 \) of the ejected material on Io using the kinematic equation for maximum height: \( v_0 = \sqrt{2 \cdot g_{Io} \cdot h_{Io}} \), where \( h_{Io} = 500 \times 10^3 \text{m} \).
3. Now, calculate the gravitational acceleration on Earth, \( g_{Earth} = 9.81 \text{m/s}^2 \).
4. Use the initial velocity \( v_0 \) calculated for Io to find the maximum height \( h_{Earth} \) the material would reach on Earth using the formula: \( h_{Earth} = \frac{v_0^2}{2 \cdot g_{Earth}} \).
5. Finally, compare the maximum height \( h_{Earth} \) with \( h_{Io} \) to understand how the gravitational differences affect the ejection height on Earth.

Key Concepts

Gravitational Acceleration

Gravitational acceleration is the rate at which an object accelerates due to the force of gravity. On Earth, this is approximately 9.81 m/s², while on Io, it is significantly less due to its smaller mass and radius. Understanding gravitational acceleration is crucial for calculating the height to which ejected material can rise on different celestial bodies.

Escape Velocity

Escape velocity is the minimum speed needed for an object to break free from a celestial body's gravitational pull without further propulsion. It depends on the mass and radius of the body. For Io, the escape velocity is lower than Earth's due to its smaller mass and size, affecting how high volcanic material can be ejected.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration, such as gravity. They are used to calculate the maximum height an object reaches when projected upwards. By applying these equations, one can determine how high material ejected from Io would travel if subjected to Earth's gravitational acceleration.