Kepler's Law for finding velocity

In summary, Kepler's second law states that each planet sweeps equal areas for equal periods of time. To solve for the new velocity, one needs to use Kepler's Third Law and plug in the given answer for r_A and r_P.
  • #1
Carpetfizz
13
0

Homework Statement


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I need to help solving part a)

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Homework Equations


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$$v= \frac{2\pi}{T}$$

$$(\frac{T_1}{T_2})^2 = (\frac{r_1}{r_2})^3$$

The Attempt at a Solution



I'm not sure where to begin really. One approach I tried was getting T_1 in terms of v_0 and plugging it into Kepler's Third Law to get T_2, and plugging T_2 back into the velocity equation to get the new velocity. However, this did not match the given answer of:

$$v_{periastrom} = \frac{r_A}{r_P}v_0$$
 
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  • #2
There's only one orbit, so only one period. That rules out Kepler III as being of use here. What's left?
 
  • #3
Kepler's second law? That each planet sweeps equal areas for equal periods of time? Not sure how to use that here.
 
  • #4
Carpetfizz said:
Kepler's second law? That each planet sweeps equal areas for equal periods of time? Not sure how to use that here.
Yes. Kepler II is essentially a statement about the conservation of angular momentum. Consider a small time interval Δt and the areas swept out at the two locations over that small time interval. You might want to make a sketch using a bit of geometry to construct the approximations.
 
  • #5
Okay so I have that $$\frac{dA}{dt} = 0.5*rvsin(\theta)$$ but I don't know how to use it to solve for v. I know that $$\frac{dA}{dt}$$ is constant though.
 
  • #6
Go much simpler. Basic geometry. Draw the triangles that result assuming tangential velocities at the two locations. The areas of the triangles will approximate the areas swept out. If the Δt is small enough then these triangles will be a good approximation. In fact, as the Δt → 0, it's essentially the calculus approximation for the differential area element. What are the expressions for the areas of the two triangles?
 
  • #7
Okay, so I have this so far:

$$A = \frac{1}{2}(t_1*v_0)r_A$$
$$A = \frac{1}{2}{t_2*v}r_P$$

but since I don't know the times I can't solve for v by setting them equal to each other.
 
  • #8
Kepler II states equal areas in equal times. The times are identical.
 
  • #9
Oh okay great, thank you so much!
 
  • #10
When you write this up, use Δt to represent the time interval and make it clear that Δt is a small time interval (theoretically approaching zero). The geometric approximation using triangles to represent the areas swept out will fail if the time interval is large, as the triangle will not approximate the curvature of the ellipse.
 
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Related to Kepler's Law for finding velocity

1. What is Kepler's Law for finding velocity?

Kepler's Law for finding velocity is a mathematical principle developed by astronomer Johannes Kepler in the 17th century. It states that the velocity of a planet in its orbit around the sun is directly proportional to its distance from the sun.

2. How is Kepler's Law for finding velocity calculated?

Kepler's Law for finding velocity is calculated using the following formula: v = √(G * M / r), where v is the velocity, G is the gravitational constant, M is the mass of the sun, and r is the distance between the planet and the sun.

3. Can Kepler's Law for finding velocity be applied to objects other than planets?

Yes, Kepler's Law for finding velocity can be applied to any object in orbit around another object, as long as the mass and distance are known. This includes satellites, comets, and other celestial bodies.

4. What is the significance of Kepler's Law for finding velocity?

Kepler's Law for finding velocity is significant because it helped to prove that the planets in our solar system follow elliptical orbits around the sun, rather than circular orbits as previously believed. It also provided a mathematical understanding of the relationship between a planet's distance from the sun and its velocity.

5. Is Kepler's Law for finding velocity still used today?

Yes, Kepler's Law for finding velocity is still used today in the study of celestial objects and their orbits. It is also an important principle in the field of astrodynamics, which deals with the motion of objects in space.

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