2 months ago

## Solution 1

Guest #2181
2 months ago

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 =

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 =

Let's substitute

(1)

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

v =

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

T² = ( ) r³

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

v =

Let's find the value of the constant

=

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 =

v = 4.779 10⁴ m / s

Venus

v =

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  ( ) = 4.11 10¹¹ m

the orbital velocity of Ceres is

v =

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.

## Solution 2

Guest #2182
2 months ago
The branch of science that deals with celestial objects, space, and the physical universe as a whole.

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