Problem 46 - College Physics

Question

\(\bullet\) Volcanoes on Io. Io, a satellite of Jupiter, is the most volcanically active moon or planet in the solar system. It has volcanoes that send plumes of matter over 500 \(\mathrm{km}\) high (see the accompanying fig- ure). Due to the satellite's small mass, the acceleration due to gravity on Io is only \(1.81 \mathrm{m} / \mathrm{s}^{2},\) and \(\mathrm{Io}\) has no appreciable atmosphere. As- sume that there is no varia- tion in gravity over the distance traveled. (a) What must be the speed of material just as it leaves the volcano to reach an altitude of 500 \(\mathrm{km} ?\) (b) If the gravitational potential energy is zero at the surface, what is the potential energy for a 25 kg fragment at its maximum height on Io? How much would this gravitational potential energy be if it were at the same height above earth?

Step-by-Step Solution

Answer

(a) 1345.36 m/s; (b) 22,625,000 J on Io, 122,625,000 J on Earth.

Step 1: Understand the Physics Problem

The problem involves a projectile motion on Io, the moon of Jupiter, where volcanic matter reaches 500 km above the surface. We're tasked to find the initial velocity (speed) required for this motion and the gravitational potential energy of a 25 kg fragment at the maximum height.

Step 2: Apply Kinematic Equation for Speed

To find the initial speed needed, apply the kinematic equation: \[ v^2 = u^2 + 2as \]Where:- \( v \) is the final velocity (0 m/s at the maximum height),- \( u \) is the initial velocity (what we are looking for),- \( a \) is the acceleration due to gravity (-1.81 m/s², negative because gravity opposes the motion),- \( s \) is the displacement (500,000 meters or 500 km).Rearrange the formula to solve for \( u \):\[ u = \sqrt{v^2 - 2as} \].

Step 3: Calculate the Initial Speed

Substitute the known values into the equation:\[ u = \sqrt{0 - 2(-1.81) \times 500,000} \]\[ u = \sqrt{2 \times 1.81 \times 500,000} \]\[ u = \sqrt{1,810,000} \]\[ u \approx 1345.36 \, \mathrm{m/s} \].Thus, the speed required is approximately 1345.36 m/s.

Step 4: Find Gravitational Potential Energy on Io

The gravitational potential energy (GPE) is given by:\[ PE = mgh \]where:- \( m = 25 \) kg is the mass,- \( g = 1.81 \, \mathrm{m/s^2} \) is the gravity on Io,- \( h = 500,000 \) m is the height.Calculate:\[ PE = 25 \times 1.81 \times 500,000 \]\[ PE = 22,625,000 \, \mathrm{J} \].So, the GPE is 22,625,000 Joules.

Step 5: Calculate Gravitational Potential Energy on Earth

The gravitational potential energy at the same height above Earth, where \( g = 9.81 \, \mathrm{m/s^2} \), is:\[ PE = mgh \]Substitute the values:\[ PE = 25 \times 9.81 \times 500,000 \]\[ PE = 122,625,000 \, \mathrm{J} \].Thus, the GPE on Earth would be 122,625,000 Joules.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy (GPE) is a form of energy that an object possesses due to its position in a gravitational field. It is dependent on three factors: the object's mass, the height above the reference point, and the local gravitational force. The formula for calculating gravitational potential energy is given by \( PE = mgh \), where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity, and \( h \) is the height above the reference point.

In the exercise, we calculated the gravitational potential energy of a 25 kg fragment at 500 km above the surface of Io. We used the fact that the acceleration due to gravity on Io is 1.81 m/s². This lower gravitational pull compared to Earth means that the same mass will have less gravitational potential energy on Io than it would on Earth. On Io, the GPE calculated is 22,625,000 Joules, but on Earth, with a stronger gravitational force of 9.81 m/s², the GPE would be significantly higher, at 122,625,000 Joules. This demonstrates how a stronger gravitational force increases an object's potential energy for the same height and mass.

Initial Velocity Calculation

Initial velocity calculation is key when determining the speed needed for an object to reach a certain height under projectile motion. To find this initial velocity, we can employ one of the kinematic equations: \( v^2 = u^2 + 2as \), where \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the acceleration due to gravity, and \( s \) is the displacement.

In our scenario concerning Io's volcanic plume, the final velocity \( v \) at the peak height is 0 m/s because the object comes to a stop momentarily before falling back. By setting \( v = 0 \) and rearranging the equation to solve for \( u \), we can find the initial speed required to propel the material to a 500 km height.

• Substitute the negative acceleration value \(-1.81 \text{ m/s}^2\), as gravity opposes the launch.
• The displacement \( s \) is 500,000 meters (equivalent to 500 km).

When solved, the initial velocity \( u \) comes out to approximately 1345.36 m/s. This speed is what the fragment would need to be launched at to overcome the gravitational pull of Io and reach the desired height.

Kinematic Equations

Kinematic equations describe the motion of objects and allow us to calculate unknown parameters such as velocity, displacement, and acceleration over time. They presume constant acceleration and are pivotal in solving problems related to projectile motion.

For vertical motion under gravity, the equations provide a way to relate the initial and final velocities of a projectile, its acceleration, and displacement. In our exercise, the kinematic equation \( v^2 = u^2 + 2as \) was particularly useful. Here, it helped us determine the initial velocity needed to propel volcanic material to significant heights on Io.

• The parameter \( v \) is the final velocity at the peak (which is zero in this context).
• \( u \) is the unknown initial velocity we aim to find.
• \( a \) represents the constant acceleration due to gravity, which is \(-1.81 \text{ m/s}^2\) on Io.
• \( s \) stands for the displacement, or the height the material needs to achieve, such as 500,000 meters in our scenario.

The understanding and application of these equations are crucial for accurately predicting the characteristics of an object's motion, especially when dealing with varying gravitational forces like those found on different planetary bodies.