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Measuring the Orbital Speeds of Planets
Introduction
A Boeing 747 can fly 624 miles per hour. That’s pretty fast! Right now, Earth and other objects in the
solar system are orbiting the Sun, but how fast are these objects moving? In this assignment, you will
calculate the orbital speeds of objects in our solar system.
Part A: Planetary Periods
In the 1600s, Johannes Kepler discovered three laws that govern the motion of objects around the sun.
Kepler’s first law explains that objects travel in an elliptical, or oval-shaped, path around the sun.
Kepler’s second law explains that an object travels faster in its orbit when it is closer to the Sun and
slower in its orbit when it is farther away. In this activity, you will investigate Kepler’s third law: the
relationship between an object’s distance from the sun and the time it takes the object to complete an
orbit around the sun.
Access the Gizmo from within today's lesson. The exploration guide found within this simulation will
help you work through the Gizmo. Click on the Lesson Materials link, which appears on the top left
corner of the screen once you enter the simulation, to access the student exploration sheet. Use it as
you work through Activity C and answer all parts.
Part B: Calculating Orbital Speed
Every object moves at a different average speed in its orbit around the Sun. In this activity, you will use
data collected from Part A to calculate the average orbital speeds of each planet and of Pluto.
1. Recall that the path an object takes around the Sun is an ellipse, and not a circle. However, to
approximate the circumference, or length, of each orbit, you will treat the orbits as circles. Go
back to your notes in your student exploration sheet from Part A. Using the mean orbital radius
(R) that you recorded, calculate the circumference of each planet’s orbit and record it in the
table below. Use the following formula to calculate the circumference:
Circumference 2 = π r
2. Using the periods you recorded in your student exploration sheet and the circumference of orbit
that you just calculated, calculate the average orbital speed of each planet and record it in the
table below. Use the following formula to calculate speed:
Distance Circumference Average Orbital Speed Time Period = =
The units of your answer will be in AU/year.
3. To give you a better perspective of how fast these orbital speeds really are, convert the orbital
speeds you just calculated from AU/year to miles/hour. 1 AU/year = 10,604 miles/hour. Record
your answers in the table below.
Table
Object
Mean
orbital
radius (AU)
(copy from
#6 in part A)
Circumference of
Orbit (AU)
Period
(Earth
Years)
(copy
from #6
in part
A)
Orbital Speed
(AU/year)
Orbital Speed
(miles/hour)
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Drawing Conclusions
4. Which planet takes the longest amount of time to travel once around its orbit? Which planet
takes the shortest amount of time?
5. How does an increase in orbital radius affect average orbital speed?
Extension
perihelion

In astronomy, the term refers to the point in the orbit of an asteroid, comet, or planet at
which it is closest to the sun. Aphelion refers to the point at which the object is furthest away from the
sun. Research facts about the dwarf planet Ceres. Calculate the average orbital speed of Ceres in
miles/hour when it is closest to the sun and when it is farthest away.

Respuesta :

the branch of science that deals with celestial objects, space, and the physical universe as a whole.

Kepler's third law allows finding the answers for the orbital speed of the planets are:

  • The speed decreases as we move away from the Sun, the values ​​are in the third column of the table.
  • The fastest planet is Mercury and the slowest planet is Pluto.

Kepler measured and analyzed the astronomical data of him and his tutor Brake, finding mathematical relationships that describe the movement of the planet, they are called Kepler's laws

      1. The orbits are ellipses

      2. A vector from the sun to the planet travels equal areas in equal times

      3. A relationship between the period and the semi-major axis of the orbit.

Kepler's third law is an application of Newton's second law to the motion of the planets around the sun.

Newton's second law establishes a relationship between force and the product of mass and acceleration of the object; in this case the force is the gravitational attractive force

           F = m a

           F = [tex]G \frac{M m}{r^2}[/tex]

Wher M y m are sum and planet mass, r is the distance ang G constnate universal gravitation

They indicate that we consider the orbits as circular, in this case the acceleration is centripetal

           a = [tex]\frac{v^2}{r}[/tex]

Let's substitute

         [tex]G \frac{M m}{r^2} = m \frac{ v^2}{r}[/tex]    

         [tex]v = \sqrt{ \frac{GM }{r} }[/tex]              (1)

The modulus of velocity (speed) is constant so we can use the uniform motion ratio

           v = [tex]\frac{\Delta x}{t}[/tex]

For a complete orbit, the distance traveled is the length of the circle and the time is called the period.

           Δx = 2π r

we substitute

            [tex]\frac{G M}{r} = \frac{4 \pi^2 r^2 }{T^2}[/tex]

            T² = ( [tex]\frac{4\pi ^2 }{GM}[/tex]) r³

They ask to calculate the orbital velocity we can use the relation 1

           v = [tex]\frac{\sqrt{GM }}{ \sqrt{r} }[/tex]

           

Let's find the value of the constant

           [tex]\sqrt{GM}[/tex] = [tex]\sqrt{6.67 \ 10^{-11} \ 1.991 \ 10^{30}}[/tex]

           [tex]\sqrt{GM} = 1.15 \ 10^{10}[/tex]

In the table we have the tabulated values ​​for the radii of the orbits of the planets

planet      radius orbit (m)    velocity (m/s)

Mercury       5.79 10¹⁰              4.779 10⁴

Venus          1.08 10¹¹               3.499 10⁴

Land            1,496 10¹¹             2,973 10⁴

Mars            2.28 10¹¹              2.408 10⁴

Jupiter        7.78 10¹¹               1.304 10⁴

Saturn         1.43 10¹²                9.62 10³

Uranus        2.87 10¹²               6.79 10³

Neptune     4.50 10¹²              5.42 10³

Pluto           5.91 10¹²               4.73 10³

We calculate the speed of some as an example and the others are in the third column of the table

Mercury

            v = [tex]\frac{1.15 \ 10^{10}}{\sqrt{5.79 \ 10^{10}} }[/tex]

            v = 4.779 10⁴ m / s

Venus

            v = [tex]\frac{1.15 \ 10^{10}}{\sqrt{1.08 \ 10^{11}} }[/tex]

            v = 3,499 10⁴ m / s

They ask to know the planet that has maximum and minimum orbital speed.

After reviewing the calculations, and observed the table,   Mercury has the highest speed and Pluto is the one with the lowest orbital speed.

When examining the expression the velocity is inversely proportional to the square root of the radius of the orbit

The radius of the orbit of Ceres is r = 2.766 ua  ( [tex]\frac{1.49 \ 10^{11} m}{1 ua}[/tex]) = 4.11 10¹¹ m

the orbital velocity of Ceres is

           v = [tex]\frac{1.15 \ 10^{10}}{\sqrt{4.11 \ 10^{11}} }[/tex]

           v = 1.79 10⁴ m / s

those that correspond to a speed between Mars and Jupiter

In conclusion using Kepler's third law we can find the answers for the orbital speed of the planets are:

  • The speed decreases as we move away from the Sun, the values ​​are in the third column of the table.
  • The fastest planet is Mercury and the slowest planet is Pluto.

Learn more here:  brainly.com/question/9622816

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