# Homework Help: What is Pluto's speed at the most distant point in its orbit?

1. Jan 26, 2010

1. The problem statement, all variables and given/known data

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?

2. Relevant equations

Conservation of energy:
$$K_{2} + U_{2} = K_{1} + U_{1}$$

3. The attempt at a solution

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

$$M_{s}$$ = Mass of Sun

$$\frac{1}{2}M_{p}v_{2}^{2} - \frac{GM_{s}M_{p}}{r_{2}} = \frac{1}{2}M_{p}v_{1}^{2} - \frac{GM_{s}M_{p}}{r_{1}}$$

After fiddling with the equation above, I get:

$$v_{2} = v_{1} + \sqrt{\frac{2GM_{s}}{r_{2}-r_{1}}}$$

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

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

Perhaps a 2nd set of eyes could find where I went wrong on this.

Last edited: Jan 26, 2010
2. Jan 27, 2010

### tiny-tim

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. Jan 27, 2010

### D H

Staff Emeritus
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. Jan 27, 2010

### tiny-tim

ooh, i didn't think of that!

5. Jan 27, 2010