Using Kepler's Law to find speed of a comet in orbit

[johannaposey]

Comets travel around the sun in elliptical orbits with large eccentricities. If a comet has speed 2.5×10⁴ when at a distance of 2.3×10¹¹ from the center of the sun, what is its speed when at a distance of 5.9×10¹⁰ .
Using Kepler's Law T² is proportional to R³
T²/R³ is a constant (C)
-->T= (2*\pi*R)/v -->T²= (4*\pi²*R²)/v²

therefore
C= (4*\pi²)/(R*v²)

since
4*\pi² is a constant C= 1/(R*v²)

therefore
R₁*(v₁)²=R₂*(v₂)²

Plugging in the variables, I got 4.93x10⁴, but that answer was wrong. Why?

 

[SammyS]

Comets travel around the sun in elliptical orbits with large eccentricities. If a comet has speed 2.5×10⁴ when at a distance of 2.3×10¹¹ from the center of the sun, what is its speed when at a distance of 5.9×10¹⁰ .

Using Kepler's Law T² is proportional to R³
T²/R³ is a constant (K)
-->T= (2*\pi*R)/v -->T²= (4*\pi²*R²)/v²

therefore
K= (4*\pi²)/(R*v²)

since
4*\pi² is a constant K= 1/(R*v²)

therefore
R₁*(v₁)²=R₂*(v₂)²

Plugging in the variables, I got 4.93x10⁴, but that answer was wrong. Why?

Hello johannaposey. Welcome to PF !

Aren't there three of Kepler's Laws ?

The one you used relates the period for the whole orbit and the orbits average distance from the Sun.

You have information regarding the comet at two particular places in its orbit. Use one of the other two laws.

 

[gneill]

Also, what are the units? Furlongs and fortnights? Always be sure to specify the units!

 

[johannaposey]

Every distance is in meters. I assume I will need to use Kepler's second law (stating that a line joining the planet and the sun sweeps out equal areas during a time interval). Is this right? If it is how do I use it. I don't have the orbital velocity.

 

[gneill]

Does the problem statement specifically direct you to use Kepler's laws to to solve it? I'd be more inclined to look at conservation of energy.

 

[johannaposey]

I tried conservation of energy, but got stuck. I tried U1=K1+K2+U2 Where U1 is the potential energy of the system at the beginning (Gm1m2/r1) K1 is the kinetic energy of the smaller mass (1/2 m1v1^2) K2 is the kinetic energy of the larger mass (1/2 m2v2^2) and U2 is the potential energy at the second radius (Gm1m2/r2). After I did that tI had two unknowns v1 and v2.

 

[gneill]

I tried conservation of energy, but got stuck. I tried U1=K1+K2+U2 Where U1 is the potential energy of the system at the beginning (Gm1m2/r1) K1 is the kinetic energy of the smaller mass (1/2 m1v1^2) K2 is the kinetic energy of the larger mass (1/2 m2v2^2) and U2 is the potential energy at the second radius (Gm1m2/r2). After I did that tI had two unknowns v1 and v2.

You only need to consider the KE and PE of the comet; the Sun is taken to be a stationary object since its mass is much, much greater than that of the body in orbit. If m is the comet's mass and M the Sun's mass, then what is called the 'specific mechanical energy' for the orbit is given by
$$\xi = \frac{v^2}{2} - \frac{G M}{r}$$
and is a constant of the motion.

 

[johannaposey]

Oops! I was getting two questions confused. I don't know why I didn't try that earlier. I've got the right answer now. Thank you!

 

[SammyS]

Oops! I was getting two questions confused. I don't know why I didn't try that earlier. I've got the right answer now. Thank you!

What did you get for the answer?