Rank the following objects, in order of their circular velocities, from smallest to largest. a. a \(5-\mathrm{kg}\) object orbiting Earth halfway to the Moon b. a \(10-\mathrm{kg}\) object orbiting Earth just above Earth's surface c. a 15 -kg object orbiting Earth at the same distance as the Moon d. a 20 -kg object orbiting Earth one-quarter of the way to the Moon

Orbital mechanics is the study of the motions of artificial satellites and natural celestial objects under the influence of forces such as gravity. It explains how objects like planets, moons, and satellites move in their orbits.
For example, when a satellite orbits Earth, it must follow certain rules dictated by gravitational forces. The key concept here is that the closer an object is to Earth, the faster it must travel to stay in orbit. This is because the gravitational pull is stronger when the object is closer to Earth.
Understanding orbital mechanics helps us place satellites in the right orbits, plan space missions, and predict movements of celestial bodies.
Here are a few important points to remember:

• Velocity varies with distance: The nearer the object to the Earth, the higher its orbital velocity.
• Orbits are paths: Different objects take different paths around the Earth based on their distance and speed.

The gravitational constant, denoted as \(G\), is a key part of understanding how gravity works in our universe. It is a constant value used in the formula to calculate gravitational forces between two objects.
Newton’s Law of Universal Gravitation uses \(G\) in its formula:
\[ F = G \frac{m_1 m_2}{r^2} \]
where \( F \) is the gravitational force, \( m_1 \) and \( m_2 \) are the masses of two objects, and \( r \) is the distance between their centers.
When looking at circular velocity, we use a decoupled form that includes \(G\) to examine how fast an object must travel to stay in a stable orbit. The formula is: \(V_c = \sqrt{\frac{GM}{r}}\)
Here’s what is important to remember:

• \(G\) is constant and does not change.
• It helps calculate gravitational force, so we can understand how two masses influence each other.
• In orbital calculations, it helps determine the speed needed for an orbiting object.

Distance from Earth is crucial in determining the circular velocity of objects in orbit. This distance is often measured from the center of the Earth to the center of the orbiting object.
The greater the distance, the weaker the gravitational pull from Earth, meaning the object would travel slower in its orbit.
Let’s break down some specific distances mentioned in the exercise:

• Halfway to the Moon: This is about 0.5 times the average distance from Earth to the Moon.
• Earth’s surface: This is essentially the radius of the Earth, a key reference point.
• Same distance as the Moon: This is the full average distance from Earth to the Moon, which is about 384,400 km.
• Quarter way to the Moon: This is 0.25 times the distance to the Moon.

In summary, objects closer to Earth need to travel faster to maintain orbit due to stronger gravitational forces. As the distance increases, the required circular velocity decreases, explaining why objects farther from Earth, like the Moon, have slower orbital velocities.