How Do You Determine if a Meteoroid Has an Open Orbit?

[Workout]

Homework Statement

A meteoroid with a velocity (when the effect of the Earth's gravity has been eliminiated) of 49.9 km/s when it is at a distance of 1.50x10^11m from the sun.

(a) Using energy calculations show that this object has an open orbit and is therefore not restricted to our solar system
(b) Calculate what the velocity of this meteoroid would be when it was infinitely far away from the sun

Homework Equations

v = sqrt (2GM/r)

The Attempt at a Solution

For (a) I just plugged in v = 49 900 m/s
G = constant
r = 1.50x10^11m

And I solved for M which was 9.33x10^30 kg which doesn't equal the Earth of the mass so that shows that it isn't part of our solar system.

Then for (b) I used the same equation I used above and plugged in r = infinity which eventually leads to 0. So does the v eventually go to v = 0 m/s?

 

[tms]

The first part of the problem says to use energy calculations. So, find the meteroid's energy, and compare it to how much energy is needed to escape the Sun. Don't forget to include kinetic and potential energy.

 

[Workout]

I used the energy formula E = KE + U
= 1/2mv^2 - GmM/r
E = 1/2v^2 - GM/r

Then I plugged in my knowns
v = 49900 m/s
G = 6.67x10^-11
M = 5.98x10^24 kg
r = 1.50x10^11m

So the energy of the meteroid in the Earth's circuit should be 1.25x10^9 J

So for the comparison do I compare it to the escape velocity? Because it doesn't workout because E = 0. A little confused.

 

[rcgldr]

Here I assume E = 0 because it said that the meteroid is not in our solar system so it's escape velocity right? So then my formula provided is the energy formula for escape velocity.

You have the formula for escape velocity:

v_e = sqrt(2 G M / r)

and the velocity of the meteoroid is given:

v_m = 49 kph

so what do you think the next step should be to determine if the meteroid has achieved escape velocity?

For the last part of the problem, you could also use the formula for gravitation potential or gravitational potential energy versus distance from the sun.

 

[Workout]

Do I get the escape velocity and compare it to the velocity of the meteoroid?

 

[tms]

So for the comparison do I compare it to the escape velocity? Because it doesn't workout because E = 0. A little confused.

What would the energy of an object at that location at escape velocity be?

 

[rcgldr]

Do I get the escape velocity and compare it to the velocity of the meteoroid?

Yes.

 

[Workout]

Yes.

Thank you.

 

[tms]

Yes.

Except that the problem says to use energy calculations.

 

[rcgldr]

Except that the problem says to use energy calculations.

So that would change this to

Spoiler

1/2 m v_e² = G M m / r = - GPE of meteroid

1/2 m v_m² = 1/2 m (49900)² = KE of meteroid