Did I Calculate the Speed of a Comet Correctly?

[Albertgauss]

Homework Statement

A comet has a speed of 300 km/sec at a distance to the sun of 100 million kilometers. What is the Comet's speed when it is at the orbit of Jupiter?

Relevant Equations

Conservation of Energy for planets

Hi all,

I just want someone to double-check my calculation. Using some typical values of a comet, if the comet has a speed of 300 km/sec when it is 100 million kilometers from the sun, how fast is the comet moving when it hits the radius of Jupiter from the sun (5 Au)?

I think I did everything right, conserving energy for planets, but the result I got was surprising: the comet's speed barely changes when it is at the orbit of the Jupiter from the sun, and thus, is very constant for the inner planets. I attached a jpeg of what I did. Is everything correct I did I do something wrong? I guess 5 Au is not nearly as much for a comet as I thought it would be.

 

[phyzguy]

You did something wrong in your GM/r calculation. The potential and kinetic energy terms should have similar magnitudes. Show us the details of the values you used to calculate GM/r.
Edit: I don't think you did the kinetic term right either.

 

[Albertgauss]

I attached the calcs in more detail. Note that the mass of the comet falls out. I did make one mistake, the mass of the sun is 2(10^30) kg and not 2(10^28) kg as I had previously. Nevertheless, even with this factor of 100, the speed at the orbit of Jupiter is abut 296 km/sec, not much different from the original 300 km/sec at 100 million km of the sun.

 

[phyzguy]

OK, you found the one factor of 100 error in GM. But you keep jumping back and forth between the kinetic term being 4.5x10^9 and being 45x10^9. Which is it? I think your bottom line conclusion that the speed doesn't change much is correct. This is because 300 km/sec is a very high speed, not at all typical. A more typical speed is about 50 km/sec. The Earth's speed in its orbit is about 30 km/sec. Are you sure the problem says 300 km/sec?

 

[Albertgauss]

You're right that there is a typo in my first slide, that should be 45(10^9) also. So, yes, the final answer is 45(10^9) which comes from 300 km/sec squared, expressed as 300,000 meter per sec (then square) to get 90(10^9) and then divide by 2 from the "1/2" of the kinetic energy term.

I did use "300 km/sec" when a questionable source--I now realize---used 300 miles per second as the comet speed. I copied down "km" and not "miles" though I don't want to change that now. I admit I should have clicked on the source and read it more thoroughly. I attached a jpeg of what I got when I google searched for comet speeds. Below is the search I did for "how fast do comet's travel".

I did think that the above source was questionable but then here is a source that claims the comet can hit 800 km/sec at peri, but they don't use the conservation of energy orbit of circular motion like I did; I'm fine with that, I just wanted ballpark numbers anyway. They use the true ellipse nature of the comet, but I don't want to get too complicated if I don't have to.

The website below that comets can hit 600 km/sec when they crash into the sun. I don't know how close to the sun that comet was when it was at 600 km/sec.

https://www.space.com/33651-comet-death-dive-into-sun-video.html

There is also "Borisov", a comet from beyond the solar system, and it has values of 50 km/sec at 2 Au. This sounds more typical of what you mentioned. I also did find a table of comet speeds around 50 km/sec like you said, with Haley at 66 km/sec according to what someone calculated.

It was hard to find any organized list of comet speeds near perihelion.

 

[phyzguy]

Bodies gain speed as they get closer to the sun, because of conservation of energy, as you were calculating. So the speed is much faster near perihelion. But to still be moving at 300 km/sec at 100 million km from the sun is moving very fast. Most comets (not all) are in bound orbits, meaning that KE - PE is negative. In your example KE - PE is strongly positive, as you found. This would mean the comet entered the solar system from elesewhere at a very high rate of speed. Not impossible, but unusual.

 

[Albertgauss]

Sounds good. Anyway, I feel my question was answered. Other comments always welcome, but I got what I needed. Appreciate the help and the second pair of eyes helped me to catch lots of mistakes.