Projectile and Satellite Motion: Concepts and Applications

Projectile and Satellite Motion

Nature of Science

Scientific ideas evolve over time as new evidence and theories emerge. Newton's gravity-based worldview has been superseded by Einstein's theory of general relativity, which describes gravity as the distortion of space and time by mass. However, Newton's law of universal gravitation remains widely used for practical applications such as satellite motion.

• Newton's Law of Universal Gravitation: Describes the attractive force between any two masses.

• Einstein's Theory of General Relativity: Explains gravity as the curvature of spacetime caused by mass and energy.

• Scientific Progress: Science is not static; theories are refined or replaced as new data becomes available.

• Practical Use: Newton's laws are still used for calculations involving satellites and planetary motion.

Example: Satellite motion calculations typically use Newton's law for simplicity and accuracy in most cases.

Gravity Review

Fundamental Properties of Gravity

Gravity is a universal force that attracts all objects with mass towards each other. Its effects are observable at all scales, from people on Earth to planetary and satellite motion.

• Universal Attraction: Every object in the universe attracts every other object due to gravity.

• Mass Dependency: The gravitational force is stronger between more massive objects.

• Distance Dependency: The force of gravity decreases as the distance between objects increases.

Example: The Moon orbits the Earth due to the gravitational attraction between them.

Newton's Law of Universal Gravitation

Newton's law quantifies the gravitational force between two masses:

• Formula:

$F = G \frac{m_1 m_2}{r^2}$

• F: Gravitational force

• G: Universal gravitational constant ($6.674 \times 10^{-11} \text{N m}^2/\text{kg}^2$)

• m_1, m_2: Masses of the two objects

• r: Distance between the centers of the masses

Application: Used to calculate the force between Earth and a satellite.

Projectile Motion

Definition and Components

A projectile is any object that moves through the air or space under the influence of gravity, continuing in motion by its own inertia. Projectile motion is a combination of horizontal and vertical components.

• Horizontal Component: Remains constant (if air resistance is negligible).

• Vertical Component: Changes due to the acceleration caused by gravity.

• Parabolic Path: The trajectory of a projectile is a parabola when only gravity acts vertically.

Example: A ball thrown horizontally and a ball dropped vertically from the same height will hit the ground at the same time (if air resistance is ignored).

Equations of Projectile Motion

• Horizontal Distance:

$x = v_{0x} t$

• Vertical Position:

$y = v_{0y} t - \frac{1}{2} g t^2$

• Maximum Height:

$h_{max} = \frac{v_{0y}^2}{2g}$

• Time of Flight:

$t_{total} = \frac{2 v_{0y}}{g}$

Satellite Motion

Satellites as Projectiles

Satellites are fast-moving projectiles that fall around Earth rather than into it. They require sufficient tangential velocity to remain in orbit, and in the absence of air resistance, they can orbit indefinitely.

• Orbital Speed: For low Earth orbit, the required speed is about 8 km/s.

• Curvature Matching: The satellite's path matches the curvature of Earth, so it continually falls around the planet.

• Newton's First Law: Without gravity, a satellite would move in a straight line; gravity causes it to follow a curved path.

Example: The International Space Station orbits Earth at an altitude where air resistance is minimal but still requires periodic boosts.

Circular Satellite Orbits

In a circular orbit, the satellite moves perpendicular to the force of gravity, maintaining a constant speed and distance from the center of the Earth.

• Period of Orbit: For satellites close to Earth, the period is about 90 minutes; higher altitudes result in longer periods.

• Launch Procedure: Rockets are initially fired vertically, then tipped and accelerated horizontally to achieve orbital speed.

Escape Velocity

Escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a massive body.

• Formula:

$v_{esc} = \sqrt{\frac{2GM}{R}}$

• Earth's Escape Velocity: Approximately 11.2 km/s at the surface.

• Energy Condition: At escape velocity, the sum of kinetic and gravitational potential energy is zero.

Example: A spacecraft must reach escape velocity to leave Earth's gravitational field and travel into deep space.

Energy Conservation in Satellite Motion

Circular Orbits

In a circular orbit, the kinetic energy (KE) and potential energy (PE) of the satellite remain constant because the distance from the center of the attracting body does not change.

• Energy Conservation: $PE + KE = \text{constant}$

• No Change in Speed: The speed and kinetic energy remain unchanged.

Elliptical Orbits

In an elliptical orbit, the satellite's speed and energy vary depending on its position.

• Apogee: The point farthest from Earth; potential energy is greatest, kinetic energy is least.

• Perigee: The point closest to Earth; kinetic energy is greatest, potential energy is least.

• Energy Conservation: The sum of kinetic and potential energy remains constant throughout the orbit.

Example: Satellites in elliptical orbits speed up as they approach Earth and slow down as they move away.

Weightlessness in Space

Apparent Weightlessness

Astronauts appear weightless in orbit because they are in continuous free fall towards Earth, experiencing no normal force from a surface. The sensation is similar to a person jumping on a trampoline and feeling weightless at the peak of the jump.

• Gravity Still Acts: The force of gravity is nearly the same as at Earth's surface.

• No Support Force: Without a surface pushing up, the weight force cannot be felt.

Example: Astronauts in the International Space Station experience microgravity due to their orbital motion.

Summary Table: Key Concepts in Projectile and Satellite Motion

Concept	Definition	Key Formula	Example/Application
Gravity	Attractive force between masses	$F = G \frac{m_1 m_2}{r^2}$	Earth-Moon attraction
Projectile Motion	Motion under gravity with horizontal and vertical components	$y = v_{0y} t - \frac{1}{2} g t^2$	Thrown ball, cannonball
Satellite Orbit	Projectile with sufficient tangential velocity to fall around Earth	$v_{orbital} = \sqrt{\frac{GM}{R}}$	International Space Station
Escape Velocity	Minimum speed to leave gravitational field	$v_{esc} = \sqrt{\frac{2GM}{R}}$	Spacecraft leaving Earth
Weightlessness	Apparent lack of weight due to free fall	N/A	Astronauts in orbit

Additional info: Some equations and context have been expanded for clarity and completeness. The summary table is inferred from the main concepts presented in the notes.