Satellite Motion: Intro: Videos & Practice Problems

A satellite is any object that orbits another, such as the Moon around Earth or Earth around the Sun. The shape of a satellite's orbit depends on its speed and distance from the central mass. Newton proposed that a minimum speed, known as $v\_\mathrm{orbit}$, allows an object to maintain a circular orbit, while speeds above this lead to elliptical orbits. Escape velocity, or $v\_\mathrm{escape}$, is the speed needed to break free from gravitational pull. Understanding these concepts is crucial for satellite motion analysis.

Intro to Satellite Motion

Satellites are defined as objects that orbit around a larger mass. For example, the Moon orbits the Earth, while the Earth orbits the Sun. The motion of satellites is influenced by their speed and distance from the central mass. Understanding this motion can be traced back to Newton's thought experiment involving a cannon atop a tall tower, where varying the projectile's speed leads to different outcomes.

When a projectile is launched with zero initial velocity, it falls directly to the Earth due to gravity. If it is given some horizontal velocity, it will travel a distance before hitting the ground, demonstrating projectile motion. However, if the projectile is launched at a specific minimum speed, it can achieve a stable orbit. This speed, known as the orbital speedorbit)>, allows the object to "scrape" the surface of the Earth while continuously falling towards it, creating a circular path around the planet.

For a perfectly circular orbit, there is a specific speed called the circular velocity (v_circular). If the speed is less than this value, the orbit will be elliptical. Conversely, if the speed exceeds a certain threshold, known as the escape velocity (v_escape), the object will escape the gravitational pull of the planet and will not return.

The relationship between speed and orbit shape can be summarized as follows: speeds below the minimum orbital speed result in projectile motion, speeds between the minimum orbital speed and circular velocity yield elliptical orbits, and speeds above escape velocity lead to escape trajectories. For instance, if a projectile is launched at 1500 m/s, it will behave as a projectile. At 4000 m/s, it will follow an elliptical orbit, while at 6000 m/s, it will still be elliptical but larger. Finally, a launch speed of 15,000 m/s exceeds the escape velocity, resulting in the object escaping the planet's gravitational influence.

It is important to note that the specific values for these speeds can vary based on the height from which the projectile is launched. In most satellite motion problems, circular orbits are assumed for simplicity unless stated otherwise.

Here’s what students ask on this topic:

What is a satellite in the context of physics?

A satellite in physics is any object that orbits another object due to gravitational forces. For example, the Moon is a satellite of Earth, and Earth is a satellite of the Sun. The larger mass (e.g., Earth or Sun) is the central body, while the smaller mass (e.g., Moon or Earth) is the satellite. The motion of satellites is governed by their speed and distance from the central mass, which determines the shape of their orbit.

What factors determine the shape of a satellite's orbit?

The shape of a satellite's orbit is determined by two main factors: its speed and its distance from the central mass. If the satellite's speed is at a specific value known as the circular orbit speed (v_circular), the orbit will be perfectly circular. Speeds below this value result in elliptical orbits, while speeds above the escape velocity (v_escape) allow the satellite to break free from the gravitational pull and escape into space.

What is the minimum speed required for a satellite to maintain an orbit?

The minimum speed required for a satellite to maintain an orbit is known as the orbital velocity (v_orbit). This is the speed at which the satellite just barely scrapes the surface of the central body while continuously falling towards it due to gravity. For a perfectly circular orbit, this speed must be precisely the circular orbit speed (v_circular).

What is escape velocity and how is it calculated?

Escape velocity (v_escape) is the minimum speed needed for an object to break free from the gravitational pull of a central body and never return. It can be calculated using the formula:

$\sqrt{\frac{2\cdot G\cdot M}{r}}$

where G is the gravitational constant, M is the mass of the central body, and r is the distance from the center of the mass to the object.

How does height affect the orbital and escape velocities of a satellite?

The height of a satellite above the central body affects both its orbital and escape velocities. As the height increases, the gravitational pull decreases, which means the required speeds for both orbital and escape velocities decrease. Therefore, a satellite at a higher altitude will need a lower speed to maintain its orbit or to escape the gravitational pull compared to a satellite closer to the central body.