Problem 70 - University Physics with Modern Physics

Question

A satellite with mass 848 kg is in a circular orbit with an orbital speed of 9640 m/s around the earth. What is the new orbital speed after friction from the earth's upper atmosphere has done \(-\)7.50 \(\times\) 10\(^9\) J of work on the satellite? Does the speed increase or decrease?

Step-by-Step Solution

Answer

The new speed is 8656 m/s; the speed decreases.

Step 1: Understand the problem statement

We are asked to find the new orbital speed of a satellite after work has been done on it due to friction from the earth's atmosphere. The important information given includes the initial mass of the satellite, its initial speed, and the work done by friction.

Step 2: Apply the Work-Energy Principle

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy: \( W = \Delta KE \). The initial kinetic energy \( KE_{i} \) of the satellite can be calculated using the formula \( KE_{i} = \frac{1}{2}mv^2 \), where \( m = 848 \text{ kg} \) and \( v = 9640 \text{ m/s} \).

Step 3: Calculate Initial Kinetic Energy

Compute the initial kinetic energy:\[KE_{i} = \frac{1}{2} \times 848 \text{ kg} \times (9640 \text{ m/s})^2= 39270310400 \text{ J}.\]

Step 4: Find the New Kinetic Energy

The work done \( W = -7.50 \times 10^9 \text{ J} \). The new kinetic energy \( KE_{f} \) is computed as:\[KE_{f} = KE_{i} + W = 39270310400 \text{ J} - 7.50 \times 10^9 \text{ J} = 31770310400 \text{ J}.\]

Step 5: Calculate the New Orbital Speed

Using \( KE_{f} = \frac{1}{2}mv_{f}^2 \), solve for the new speed \( v_{f} \):\[31770310400 = \frac{1}{2} \times 848 \times v_{f}^2 \v_{f}^2 = \frac{31770310400}{424} \v_{f} \approx 8656.0 \text{ m/s}.\]

Step 6: Determine Change in Speed

Compare the initial and final speed: The initial speed was 9640 m/s and the new speed is approximately 8656 m/s. This shows a decrease in speed.

Key Concepts

Work-Energy Principle

The work-energy principle is fundamental in physics and applies especially well to orbital mechanics. Essentially, this principle states that work done on an object results in a change in its kinetic energy. In formula terms, it is expressed as:

• Work Done (\( W \)) = Change in Kinetic Energy (\( \Delta KE \))

When dealing with satellites, this principle helps us understand how forces, like friction from an atmosphere, alter a satellite's motion.
For the given exercise, the friction from the Earth's upper atmosphere has done negative work on the satellite, indicating it has removed energy from the system. This reduction in kinetic energy directly influences the satellite's velocity around the Earth.
Through the calculation:

• Initial Kinetic Energy (\( KE_i \)) is calculated using: \( KE_i = \frac{1}{2}mv^2 \)
• With the work done being negative, compute the new Kinetic Energy (\( KE_f \)) as: \( KE_f = KE_i + W \)

This gives us a new kinetic energy reflecting the decreased velocity of the satellite.

Orbital Speed

Orbital speed refers to the constant speed at which a satellite travels in its orbit. It is crucial for maintaining a stable orbit and overcoming gravitational pull without falling back to Earth or escaping into space.
The orbital speed is determined by the balance between gravitational forces and the inertia of the object moving through space.

• The initial orbital speed in the exercise is 9640 m/s.
• After the work done by friction, the new speed is approximately 8656 m/s.

This speed change indicates how energy influences motion even in the vacuum of space. Let's break it down further:

• The decrease in speed suggests that energy has been lost due to frictional forces acting opposite to the direction of motion.
• A stable orbit requires sufficient speed to counterbalance Earth's gravity, ensuring the satellite doesn’t spiral downwards.

Orbital speed is a delicate balance in maintaining the satellite's intended path.

Satellite Dynamics

Satellite dynamics involve understanding how satellites move and behave in their orbits. The dynamics of a satellite are driven by various forces and energies at play. Key factors include gravitational forces, atmospheric drag, and orbital maneuvers:

• Gravitational Force: It keeps the satellite in orbit. This force is continually exerted towards the center of the Earth.
• Atmospheric Drag: Low earth orbit satellites experience drag due to remnants of the Earth's atmosphere, which can slow them down, as seen in the exercise.
• Orbital Maneuvers: Adjustments made to correct or change the satellite's orbit, often done via thrusters.

In our particular exercise, the satellite's dynamics are primarily influenced by atmospheric drag, which reduces its speed and thereby its kinetic energy. Understanding these elements helps in designing effective missions and ensuring the longevity of satellite operations.