Problem 68 - University Physics with Modern Physics

Question

A rocket with mass 5.00 \(\times\) 10\(^3\) kg is in a circular orbit of radius 7.20 \(\times\) 10\(^6\) m around the earth. The rocket's engines fire for a period of time to increase that radius to 8.80 \(\times\) 10\(^6\) m, with the orbit again circular. (a) What is the change in the rocket's kinetic energy? Does the kinetic energy increase or decrease? (b) What is the change in the rocket's gravitational potential energy? Does the potential energy increase or decrease? (c) How much work is done by the rocket engines in changing the orbital radius?

Step-by-Step Solution

Answer

The kinetic energy decreases, potential energy increases, and work done by engines is positive.

Step 1: Analyze Given Data

We are given the mass of the rocket, which is \( m = 5.00 \times 10^3 \text{ kg} \). The initial radius of the circular orbit is \( r_1 = 7.20 \times 10^6 \text{ m} \), and the final radius is \( r_2 = 8.80 \times 10^6 \text{ m} \). We need to calculate the changes in kinetic and gravitational potential energies and then find the work done by the rocket engines.

Step 2: Calculate Initial and Final Kinetic Energies

The kinetic energy of an object in a circular orbit is given by the formula \( KE = \frac{1}{2} m v^2 \). However, for an object in orbit, we also have \( v^2 = \frac{GM}{r} \) where \( G \) is the gravitational constant \( 6.674 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2 \), and \( M \) is the mass of the Earth \( 5.972 \times 10^{24} \text{ kg} \). Thus, the initial kinetic energy \( KE_1 = \frac{1}{2} m \frac{GM}{r_1} \) and the final kinetic energy \( KE_2 = \frac{1}{2} m \frac{GM}{r_2} \). Calculate both \( KE_1 \) and \( KE_2 \) and find the change \( \Delta KE = KE_2 - KE_1 \).

Step 3: Calculate Initial and Final Potential Energies

The gravitational potential energy is given by \( U = -\frac{GMm}{r} \). Calculate the initial potential energy \( U_1 = -\frac{GMm}{r_1} \) and the final potential energy \( U_2 = -\frac{GMm}{r_2} \). The change in potential energy is \( \Delta U = U_2 - U_1 \).

Step 4: Determine Whether Energies Increase or Decrease

Since \( r_2 > r_1 \), \( \frac{1}{r_2} < \frac{1}{r_1} \) and thus \( KE_2 < KE_1 \). This means the kinetic energy decreases. Similarly, \( U_2 > U_1 \), meaning the potential energy increases, since the potential energy becomes less negative.

Step 5: Calculate Work Done

The work done by the rocket engines is equal to the change in the mechanical energy of the system. This is given by \( W = \Delta KE + \Delta U \). Use the previously calculated change in kinetic and potential energies to find \( W \).

Key Concepts

Orbital Mechanics

Orbital mechanics is the study of the motion of objects in space, particularly around celestial bodies like planets. It uses principles from physics to explain how objects like rockets and satellites move. When a rocket or satellite orbits a planet, it follows a path that is determined by the gravitational pull of the planet and the object's velocity.

According to Kepler's laws and Newton's laws of motion, an object in orbit will travel in an elliptical path, but in special cases, like a circular orbit, the path can be a perfect circle. The strength of the gravitational force and the speed of the object determine the shape and stability of the orbit.

• Gravitational force keeps the object in orbit by pulling it towards the celestial body.
• Velocity keeps the object from crashing into the celestial body by moving it forward.

Understanding these mechanics is crucial to calculating how spacecraft, like rockets, can change orbit or stay in a stable orbit.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. In physics, it is expressed with the formula: \[ KE = \frac{1}{2} m v^2 \] where \( m \) is the mass of the object and \( v \) is its velocity.

In terms of orbital motion, an object's kinetic energy depends on how fast it is moving along its orbit. It plays a significant role when analyzing how a rocket or satellite's energy changes when it moves within different orbits. In a circular orbit, the speed of an object can be derived from the gravitational relation to the radius: \[ v^2 = \frac{GM}{r} \] where \( G \) is the gravitational constant and \( M \) is the mass of the celestial body being orbited.

• A smaller radius (closer orbit) means higher velocity, resulting in greater kinetic energy.
• A larger radius (farther orbit) means lower velocity, resulting in lesser kinetic energy.

This relationship means that as a rocket or satellite moves to a higher orbit, its kinetic energy decreases due to a reduction in speed.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses because of its position in a gravitational field. For objects in orbit, this energy is affected by the distance from the center of the Earth. The formula for gravitational potential energy is: \[ U = -\frac{GMm}{r} \] where:

• \( G \) is the universal gravitational constant
• \( M \) is the mass of the Earth
• \( m \) is the mass of the object
• \( r \) is the distance from the center of the Earth

Gravitational potential energy is negative because it represents a bound state within the gravitational field.

As the radius increases (the object moves to a higher orbit), the magnitude of this negative potential energy decreases (becomes less negative), suggesting an increase in potential energy. This increase occurs because more energy must be invested to move the object further from the center of the Earth.

Satellite Motion

Satellite motion involves understanding how artificial satellites orbit planets. This is important for everything from GPS functionality to telecommunications. The balance between gravitational forces and velocity enables satellites to maintain their path around a planet effectively.

For a satellite in a stable orbit, the centrifugal force due to its velocity counteracts the gravitational pull from the planet. This results in a dynamic balance that keeps the satellite from either falling back to Earth or shooting off into space.

• To change an orbit, energy must be added or removed, adjusting either speed or altitude (or both).
• If a satellite speeds up, it will move to a higher orbit.
• Slowing down will cause the satellite to descend to a lower orbit.

Understanding satellite motion helps in determining how much energy (work) is needed when changing orbits, such as when a rocket's engines fire to push it into a higher orbit as in the given exercise, resulting in a change in both kinetic and potential energies.