What is Pluto's speed at the most distant point in its orbit?

In summary, the homework statement is that Pluto's speed at its closest approach of 4.43x109km is 6.12km/s. However, at the most distant point in its orbit, where it is 7.30x109km from the sun, its speed is 3.71km/s.
  • #1
JJBladester
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Homework Statement



Pluto moves in a fairly elliptical orbit around the sun. Pluto's speed at its closest approach of 4.43x109km is 6.12km/s.

What is Pluto's speed at the most distant point in its orbit, where it is 7.30x109km from the sun?

Homework Equations



Conservation of energy:
[tex]K_{2} + U_{2} = K_{1} + U_{1}[/tex]

The Attempt at a Solution



[tex]M_{p}[/tex] = Mass of Pluto (actually cancels out when re-arranging the eq to get [tex]v_{2}[/tex])

[tex]M_{s}[/tex] = Mass of Sun

[tex]\frac{1}{2}M_{p}v_{2}^{2} - \frac{GM_{s}M_{p}}{r_{2}} = \frac{1}{2}M_{p}v_{1}^{2} - \frac{GM_{s}M_{p}}{r_{1}}[/tex]

After fiddling with the equation above, I get:

[tex]v_{2} = v_{1} + \sqrt{\frac{2GM_{s}}{r_{2}-r_{1}}}[/tex]

The correct answer is 3.71 km/s, but my answer comes out differently. I'm using the following numbers in the equation above:

[tex]v_{2} = 6.12\cdot10^{3}m/s + \sqrt{\frac{2(6.67\cdot10^{-11})(1.99\cdot10^{30})}{7.30\cdot10^{12} - 4.43\cdot10^{12}}} = 15.7km/s[/tex]

Perhaps a 2nd set of eyes could find where I went wrong on this.
 
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  • #2
Hi JJBladester! :smile:
JJBladester said:
[tex]\frac{1}{2}M_{p}v_{2}^{2} - \frac{GM_{s}M_{p}}{r_{2}} = \frac{1}{2}M_{p}v_{1}^{2} - \frac{GM_{s}M_{p}}{r_{1}}[/tex]

After fiddling with the equation above, I get:

[tex]v_{2} = v_{1} + \sqrt{\frac{2GM_{s}}{r_{2}-r_{1}}}[/tex]

I'm sorry, but this is wrong in two ways …

1/r1 - 1/r2 is not 1/(r1 - r2)

and v1 - v2 is not √(v12 - v22)
 
  • #3
Conservation of energy will work, but conservation of angular momentum will be much easier for this problem.

Note that in general, conservation of angular momentum can only tell you about the component of velocity normal to the radial vector. However, at perihelion and apohelion, the velocity vector is normal to the radial vector.
 
  • #4
D H said:
Conservation of energy will work, but conservation of angular momentum will be much easier for this problem.

ooh, i didn't think of that! :redface:
 
  • #5
Thanks all... Got it figured out now. I guess trying to do physics too late at night is a bad idea... Too many easy math errors can be made. Thanks for the good replies. That's why I love this forum... because the help is just as good as if I were taking an on-campus course.
 
  • #6
D H said:
Conservation of energy will work, but conservation of angular momentum will be much easier for this problem.

Note that in general, conservation of angular momentum can only tell you about the component of velocity normal to the radial vector. However, at perihelion and apohelion, the velocity vector is normal to the radial vector.

Wow... Great observation! The problem took under 5 minutes to solve this way and I didn't have to spend a bunch of time converting km's to m's due to the G constant being in a different set of units. Thanks again.
 

1. What is Pluto's speed at the most distant point in its orbit?

Pluto's speed at the most distant point in its orbit, also known as aphelion, varies depending on its position in its orbit. On average, it travels at a speed of about 4.74 kilometers per second.

2. How does Pluto's speed at its most distant point compare to its speed at its closest point?

Pluto's speed at its most distant point is significantly slower than its speed at its closest point, also known as perihelion. At perihelion, Pluto can reach speeds of up to 5.45 kilometers per second.

3. Does Pluto's speed at its most distant point affect its orbit?

Yes, Pluto's speed at its most distant point does affect its orbit. As Pluto moves farther away from the sun, its speed decreases, causing its orbit to become more elliptical.

4. How does the distance of Pluto's most distant point from the sun affect its speed?

The distance of Pluto's most distant point from the sun directly affects its speed. The farther away it is from the sun, the slower its speed will be.

5. How long does it take for Pluto to reach its most distant point in its orbit?

Pluto takes approximately 247.94 years to complete one orbit around the sun, which means it reaches its most distant point every 123.97 years.

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