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Escaping Our Solar System 
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#1
Jan2913, 07:29 AM

P: 937

1. The problem statement, all variables and given/known data
(a) What is the minimum speed, relative to the Sun, necessary for a spacecraft to escape the solar system if it starts at the Earth's orbit? (a) What is the minimum speed, relative to the Sun, necessary for a spacecraft to escape the solar system if it starts at the Earth's orbit? 2. Relevant equations 3. The attempt at a solution I am not exactly sure how to solve this problem. Should I use an energy approach? Do I need to calculate the escape speed for each planet, because it will pass by each one, and then sum all of the velocities together? 


#2
Jan2913, 07:40 AM

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P: 5,680

What does "escape" really mean?
Regarding your solution, consider, for example, a trajectory perpendicular to the plane of the ecliptic. Would it pass by each planet? Secondly, consider that your speed is that sufficient to escape Jupiter; would passing by Neptune need any more speed? 


#3
Jan2913, 07:52 AM

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P: 6,118

The answer is posted online. I remember looking it up. Do you know how to use Google search?



#4
Jan2913, 07:56 AM

P: 937

Escaping Our Solar System



#5
Jan2913, 08:06 AM

P: 937

EDIT: Forgot to post picture. 


#6
Jan2913, 08:22 AM

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P: 5,680

Yes, the red line would be one perpendicular to the plane of the ecliptic.
You have not said what you think "escape" means. Let's get this clear before we talk about anything else. 


#7
Jan2913, 08:28 AM

P: 937

From my understanding, escape speed is the speed necessary to reach a point away from the planet where the gravitational pull is so small it is assumed to be zero, implying that gravitational potential energy is also zero. It is at an "infinite" distance. Does that seem correct?
Oh, so the red line is correct. So, that's why we only have to consider the escape speed of the sun and earth, because it doesn't have to pass by the other planets; and because the projectile is at an "infinite" distance from earth's adjacent planets, that their gravitational pulls are negligible? 


#8
Jan2913, 08:43 AM

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P: 5,680

Anyway, "escape" means "get infinitely away" and "reach zero potential energy". What about the projectile's speed and the kinetic energy "at the infinity"? 


#9
Jan2913, 08:47 AM

P: 937

Yes, but what happens to Newton's Law of Gravitation as you go "infinitely" far? It goes to zero. In fact, it goes to zero more quickly than the Gravitational Potential Energy Function.



#10
Jan2913, 08:54 AM

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P: 5,680




#11
Jan2913, 08:57 AM

P: 937

It allows me to make the reasonable assumption that, because all of the other planets are far enough away, we don't have to consider all of them. Although the sun is far away, its mass is still large enough for it to affect the projectiles escape speed.



#12
Jan2913, 09:11 AM

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P: 5,680

I do not see any physical basis for this assumption. "Far enough away" and "mass large enough" sound very much like wishful thinking.
Forget about the entire solar system for a minute. How does one determine the escape speed for just one gravitating body? 


#13
Jan2913, 09:20 AM

P: 937

Then how else would you explain the solution. By the way, I found it on the internet. To find the escape speed in this particular case, you sum the escape speed of the sun and earth. We factor in the gravitational pull of the sun, even though we are launching from, yet we don't factor in the gravitational pull Venus or Mars. Why, because they are far enough away and have small enough mass, that there gravitational pull doesn't affect the escape speed. I am sure you could show mathematically, giving us a basis for our assumption, that the magnitude of Venus and Mars when compared to the Sun and earth would be insignificant in comparison. That's the only way I can see how to explain the solution.



#14
Jan2913, 09:24 AM

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P: 5,680

Again, I recommend that you start with just one body (a planet or the Sun). Understand how the mathematical formula for the escape speed is obtained in this simplest case. Then think how you could treat the entire solar system.



#15
Jan2913, 10:41 AM

P: 937

Now that I relook at this solution for part (a), it doesn't seem correct. http://answers.yahoo.com/question/in...3234313AAPfdJt What do you think you? Why is it wrong?



#16
Jan2913, 10:48 AM

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P: 5,680

I would like you to work out the solution for the simple problem (projectile escaping one body) first. Then we can discuss the more complex problem.



#17
Jan2913, 10:55 AM

P: 937

Well, you'd use the formula [itex]v_e = \sqrt{\frac{2GM}{r}}[/itex], where M is the mass of the object the projectile is trying to escape, and r is the radius of that object.



#18
Jan2913, 11:08 AM

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P: 5,680

That is not correct. r is not the radius, it is something else. Do you understand how this formula is obtained?



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