Physics I

Universal GravitationSatellite Motion

Special Relativity – Space & Time

Learning Objectives

• 13-Universal Gravitation & 14-Satellite Motion– Develop logical connections of Kepler’s &

Newton’s Laws to effects of physical relationships of objects moving in space.

– Describe how gravitational force between two objects depends on their masses & distance between them.

Learning Objectives

• 15-Special Relativity– Recognize that:• Nothing travels faster than light which is the same for

all observers no matter how they or the light source are moving.• Newton’s Laws are a limiting case of Einstein’s Special

Theory of Relativity at speeds that are much smaller than the speed of light.• Time, length, and energy depend on the frame of

reference.

Universal Gravitation Kepler’s Laws

Describe the motion of every planet and satellite

Kepler’s First Law

• “The paths of the planets are ellipses with the sun at one focus.

Kepler’s 2nd Law

• “An imaginary line from the Sun to a planet sweeps out equal areas in equal time intervals.

Kepler’s 3rd Law

• The square of the ratio of the periods of any two planets is equal to the cube of the ratio of their average distances from the Sun.

• Ta 2 = ra 3

• Tb rb

Law of Universal Gravitation

F = G m1 m2

d2

G = 6.67 x 10-11 Nm2/kg2

The gravitational force between any two objects is directly proportional to the product of their masses & inversely proportional to the square of the distance between their centers. The force is attractive and along a line connecting their centers.

Period of a planet orbiting the Sun.

T = 2 p r 3

G ms

The period a planet orbiting the Sun is equal to 2 p times the square root of the orbital radius cubed, divided by the product of the universal gravitational constant and the mass of the Sun.

Mass of Sun is 1.99 x 10 30 kg

The path of an Earth satellite follows the curvature of the Earth.

Chapter 14 Satellites

Throw a stone at any speed and one second later it will have fallen 5 m below where it would have been without gravity.

14.1 Earth Satellites

In the curvature of Earth, the surface drops a vertical distance of nearly 5 meters for every 8000 meters tangent to its surface.

14.1 Earth Satellites

The orbital speed for close orbit about Earth is 8 km/s.• That is an impressive 29,000 km/h (or 18,000

mi/h). • At that speed, atmospheric friction would burn

an object to a crisp. • A satellite must stay 150 kilometers or more

above Earth’s surface—to keep from burning due to the friction.

14.1 Earth Satellites

A satellite in circular orbit around Earth is always moving perpendicular to gravity and parallel to Earth’s surface at constant speed.

14.2 Circular Orbits

The satellite is always moving at a right angle (perpendicular) to the force of gravity and parallel to Earth’s surface at constant speed.

• It doesn’t move in the direction of gravity, which would increase its speed.

• It doesn’t move in a direction against gravity, which would decrease its speed.

• No change in speed occurs—only a change in direction.

14.2 Circular Orbits

For a satellite close to Earth, the time for a complete orbit around Earth, its period, is about 90 minutes. For higher altitudes, the orbital speed is less and the period is longer.

Communications satellites are located in orbit 6.5 Earth radii from Earth’s center, so that their period is 24 hours (geosynchronous). This period matches Earth’s daily rotation. They orbit in the plane of Earth’s equator and they are always above the same place. The moon is farther away, and has a 27.3-day period.

14.2 Circular Orbits

The International Space Station (ISS) orbits at 360 kilometers above Earth’s surface.

• Acceleration toward Earth is somewhat less than 1 g because of altitude.

• This acceleration, however, is not sensed by the astronauts; relative to the station, they experience zero g.

14.2 Circular Orbits

An ellipse is the closed path taken by a point that moves in such a way that the sum of its distances from two fixed points is constant. The two fixed points in an ellipse are called foci. For a satellite orbiting a planet, the center of the planet is at one focus and the other focus could be inside or outside the planet.

14.3 Elliptical Orbits

Kepler’s first law states that the path of each planet around the sun is an ellipse with the sun at one focus.

14.5 Kepler’s Laws of Planetary Motion

Kepler’s second law states that each planet moves so that an imaginary line drawn from the sun to any planet sweeps out equal areas of space in equal time intervals.

Kepler’s third law states that the square of the orbital period of a planet is directly proportional to the cube of the average distance of the planet from the sun.

Kepler’s Third Law

After ten years of searching for a connection between the time it takes a planet to orbit the sun and its distance from the sun, Kepler discovered a third law. Kepler found that the square of any planet’s period (T) is directly proportional to the cube of its average orbital radius (r).

14.5 Kepler’s Laws of Planetary Motion

Kepler was familiar with Galileo’s concepts of inertia and accelerated motion, but he failed to apply them to his own work. Like Aristotle, he thought that the force on a moving body would be in the same direction as the body’s motion. Kepler never appreciated the concept of inertia. Galileo, on the other hand, never appreciated Kepler’s work and held to his conviction that the planets move in circles.

14.5 Kepler’s Laws of Planetary Motion

Earth

Neglecting air resistance, fire anything at any speed greater than 11.2 km/s, and it will leave Earth, going more and more slowly, but never stopping.

14.6 Escape Speed

How much work is required to move a payload against the force of Earth’s gravity to a distance very, very far (“infinitely far”) away?

• Gravity diminishes rapidly with distance due to the inverse-square law.

• Most of the work done in launching a rocket occurs near Earth.

Motion through space is related to motion in time.

Chapter 15

Motion through space is related to motion in time. The first person to understand the relationship between space and time was Albert Einstein. Einstein stated in 1905 that in moving through space we also change our rate of proceeding into the future—time itself is altered. His theories changed the way scientists view the workings of the universe.

From the viewpoint of special relativity, you travel through a combination of space and time. You travel through space-time.

15.1 Space-Time

Newton and other investigators before Einstein thought of space as an infinite expanse in which all things exist. Einstein theorized both space and time exist only within the universe. There is no time or space “outside.” Einstein reasoned that space and time are two parts of one whole called space-time.

Einstein’s special theory of relativity describes how time is affected by motion in space at constant velocity, and how mass and energy are related.

You are moving through time at the rate of 24 hours per day. This is only half the story. To get the other half, convert your thinking from “moving through time” to “moving through space-time.”

• When you stand still, all your traveling is through time.

• When you move a bit, then some of your travel is through space and most of it is still through time.

15.1 Space-Time

• If you were able to travel at the speed of light, all your traveling would be through space, with no travel through time!

• Light travels through space only and is timeless. • From the frame of reference of a photon traveling

from one part of the universe to another, the journey takes no time at all!

15.1 Space-Time

Whenever we move through space, we, to some degree, alter our rate of moving into the future. This is known as time dilation, or the stretching of time. The special theory of relativity that Einstein developed rests on two fundamental assumptions, or postulates.

15.1 Space-Time

The first postulate of special relativity states that all the laws of nature are the same in all uniformly moving frames of reference.

15.2 The First Postulate of Special Relativity

Einstein reasoned all motion is relative and all frames of reference are arbitrary. A spaceship, for example, cannot measure its speed relative to empty space, but only relative to other objects.

Spaceman A considers himself at rest and sees spacewoman B pass by, while spacewoman B considers herself at rest and sees spaceman A pass by. Spaceman A and spacewoman B will both observe only the relative motion.

15.2 The First Postulate of Special Relativity

Einstein’s first postulate of special relativity assumes our inability to detect a state of uniform motion.Many experiments can detect accelerated motion, but none can, according to Einstein, detect the state of uniform motion.

15.2 The First Postulate of Special Relativity

The second postulate of special relativity states that the speed of light in empty space will always have the same value regardless of the motion of the source or the motion of the observer.

15.3 The Second Postulate of Special Relativity

Einstein asked: “What would a light beam look like if you traveled along beside it?” In classical physics, the beam would be at rest to such an observer. Einstein became convinced that this was impossible.

The speed of light is constant regardless of the speed of the flashlight or observer.

15.3 The Second Postulate of Special Relativity

Everyone who measures the speed of light will get the same value, c.

Time dilation occurs ever so slightly for everyday speeds, but significantly for speeds approaching the speed of light.

15.4 Time Dilation

Einstein proposed that time can be stretched depending on the motion between the observer and the events being observed. The stretching of time is time dilation.

Einstein showed the relation between the time t0 in the observer’s own frame of reference and the relative time t measured in another frame of reference is:

where v represents the relative velocity between the observer and the observed and c is the speed of light.

15.4 Time Dilation

The amounts of energy required to propel spaceships to relativistic speeds are billions of times the energy used to put the space shuttles into orbit.

15.5 Space and Time Travel

Before the theory of special relativity was introduced, it was argued that humans would never be able to venture to the stars.

• Our life span is too short to cover such great distances.

• Alpha Centauri is the nearest star to Earth, after the sun, and it is 4 light-years away.

• A round trip even at the speed of light would require 8 years.

• The center of our galaxy is some 30,000 light-years away, so it was reasoned that a person traveling even at the speed of light would have to survive for 30,000 years to make such a voyage!

15.5 Space and Time Travel

A person’s heart beats to the rhythm of the realm of time it is in.

• Astronauts traveling at 99% the speed of light could go to the star Procyon (11.4 light-years distant) and back in 23.0 years in Earth time.

• Because of time dilation, it would seem that only 3 years had gone by for the astronauts.

• It would be the space officials greeting them on their return who would be 23 years older.

15.5 Space and Time Travel

At higher speeds, the results are even more impressive. At a speed of 99.99% the speed of light, travelers could travel slightly more than 70 light-years in a single year of their own time. At 99.999% the speed of light, this distance would be pushed appreciably farther than 200 years. A 5-year trip for them would take them farther than light travels in 1000 Earth-time years.

15.5 Space and Time Travel

When an object moves at a very high speed relative to an observer, its measured length in the direction of motion is contracted.

15.6 Length Contraction

For moving objects, space as well as time undergoes changes.The observable shortening of objects moving at speeds approaching the speed of light is length contraction. The amount of contraction is related to the amount of time dilation. For everyday speeds, the amount of contraction is much too small to be measured.

For relativistic speeds, the contraction would be noticeable. At 87% of c, it would appear to you to be 0.5 meter long. At 99.5% of c, it would appear to you to be 0.1 meter long.As relative speed gets closer and closer to the speed of light, the measured lengths of objects contract closer and closer to zero.The width of a stick, perpendicular to the direction of travel, doesn’t change.

15.6 Length Contraction

A meter stick traveling at 87% the speed of light relative to an observer would be measured as only half as long.

15.6 Length Contraction

As relative speed increases, contraction in the direction of motion increases. Lengths in the perpendicular direction do not change.

The contraction of speeding objects is the contraction of space itself. Space contracts in only one direction, the direction of motion. Lengths along the direction perpendicular to this motion are the same in the two frames of reference.

The contraction of speeding objects is the contraction of space itself. Space contracts in only one direction, the direction of motion. Lengths along the direction perpendicular to this motion are the same in the two frames of reference.

Relativistic length contraction is stated mathematically:

v is the speed of the object relative to the observerc is the speed of lightL is the length of the moving object as measured by the observerL0 is the measured length of the object at rest

Suppose that an object is at rest, so that v = 0. When 0 is substituted for v in the equation, we find L = L0. When 0.87c is substituted for v in the equation, we find L = 0.5L0. Or when 0.995c is substituted for v, we find L = 0.1L0.

15.6 Length Contraction