# Demonstrating Kepler's Second Law - Equal Areas Swept Out in Equal Times

• UrbanXrisis
In summary: This is why an object at rest in space will stay at rest--there is no force acting on it to keep it moving. So the book is asking you to show that the shaded areas are the same, and that you have used Kepler's second law to prove it.
UrbanXrisis

I know that an imaginary line adjoining a planet and a sun sweeps out an equal area of space in equal amounts of time.

that means... I know that

$$\frac{1}{2}b(t_2-t_1)=\frac{1}{2}b(t_4-3_1)$$

but I don't know that the question means when it asks to demonstrate that the shaded areas are the same?

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UrbanXrisis said:
that means... I know that

$$\frac{1}{2}b(t_2-t_1)=\frac{1}{2}b(t_4-3_1)$$

but I don't know that the question means when it asks to demonstrate that the shaded areas are the same?

Make sure that when you're calculating the area that it results in units of distance squared (what is the relationship between the time interval, the distance travelled, and the velocity?). Your equation above has units of distance times time.

$$(A_{[1,2]} = A_{[3,4]})$$ $$\leftrightarrow$$ $$(A_{[1,2]} - A_{[3,4]} = 0)$$

So is the latter statement true? Prove it.

And like SpaceTiger wrote, when calculating those areas, don't forget, well, your velocity.

hmm... all I got is...

$$d=v\Delta t$$
$$\sqrt{b^2+(t_2-t_1)^2}=V_o (t_2-t_1)$$

not sure what to do next...

well,I see what you are saying now.

$$d=V_o (t_2-t_1)$$
$$A=\frac{1}{2} b V_o(t_2-t_1)$$

this is in terms of m^2

but what does the book mean when it says that I have to deomstrate that the shaded triangles have the same area?

UrbanXrisis said:
well,I see what you are saying now.

$$d=V_o (t_2-t_1)$$
$$A=\frac{1}{2} b V_o(t_2-t_1)$$

this is in terms of m^2

but what does the book mean when it says that I have to deomstrate that the shaded triangles have the same area?
The book is requesting you prove GEOMETRICALLY that the 2 triangles have equal areas.
Hint: (Triangle Area) = (1/2)*(Altitude)*(Base) for each triangle. Formalize what you've already done.

~~

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It looks like you're most of the way there. Incidentally, the reason Kepler's second law applies to this situation is that it represents the limit of a classical unbound orbit as the moving particle's energy goes to infinity (for an arbitrary angular momentum). In other words, the moving particle is going so fast that there is no noticable gravitational deviation about point O.

## 1. What is Kepler's Second Law?

Kepler's Second Law, also known as the Law of Equal Areas, states that the line connecting a planet to the sun sweeps out equal areas in equal amounts of time. This means that a planet will move faster when it is closer to the sun and slower when it is farther away.

## 2. How was Kepler's Second Law discovered?

Kepler's Second Law was discovered by the German astronomer Johannes Kepler in the early 17th century. He mathematically derived the law from his observations of the motion of planets, particularly Mars.

## 3. How is Kepler's Second Law demonstrated?

Kepler's Second Law can be demonstrated through a simple experiment using a string, a pencil, and a circular object such as a plate or a CD. The string is tied to the pencil at one end and the circular object at the other end. When the pencil is moved around the circular object, it will sweep out equal areas in equal amounts of time.

## 4. Why is Kepler's Second Law important?

Kepler's Second Law is important because it helps us understand the motion of planets in our solar system. It also serves as evidence for the validity of the heliocentric model of the solar system, which states that the sun is at the center and the planets orbit around it.

## 5. How does Kepler's Second Law relate to other laws of motion?

Kepler's Second Law is related to Newton's Second Law of Motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In the case of a planet orbiting around the sun, the force of gravity is acting as the net force, causing the planet to accelerate and sweep out equal areas in equal amounts of time.

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