Gravitation and its effect on elliptical orbits

[marekkpie]

Homework Statement

Comets travel around the sun in elliptical orbits with large eccentricities. Suppose the comet has an initial speed of 1.17*10^4 m/s when at a distance of 4.9*10^11 m from the center of the sun, what is its speed when at a distance of 5.9*10^10 m? Give your in m/s in scientific notation to three significant digits. (Note: Use appendix F for the necessary data.)

Homework Equations

Don't know 'em. All I know is finding velocity in a circular orbit:

v = sqrt((Gravitational constant * mass of the object being orbited around) / radius from the center of the orbited object)

The Attempt at a Solution

v = sqrt((6.67E-11 * 1.99E30) / 5.9E10) = 4.74E4

The correct answer is: 6.40E4...so since I'm in the same order of magnitude I assume I'm close.

 

[mgb_phys]

Orbits sweep out equal areas in equal times
Consider the area of a triangle swept by the comet in 1 sec (eg some time short enough that you can make the path a straight line) when it's far and close to the sun.

If the area is the same and you know the height you cna work out how long the base is and so the speed.

 

[marekkpie]

O.K...so since the times and areas are equal, we can safely set both sides to equal the area of a triangle? i.e.

.5 * R1 * V1 = .5 * R2 * V2
thus,
V2 = V1 * R1 / R2

But then I get 9.72E4.

 

[davieddy]

Orbits sweep out equal areas in equal times
Consider the area of a triangle swept by the comet in 1 sec (eg some time short enough that you can make the path a straight line) when it's far and close to the sun.

If the area is the same and you know the height you cna work out how long the base is and so the speed.

There was no mention of
perihelion or aphelion.

Conservation of energy is what is expected

David

 

[marekkpie]

Alright, then here is what I got from davieddy's help:

K + U = K + U

MC = mass of the comet

MS = mass of the sun

G = gravitational constant

So...here is my equation, followed by the steps to make it solve for V2:

.5 * MC * V1 ^ 2 - G * MS * MC / D1 = .5 * MC * V2 ^ 2 - G * MS * MC / D2
<=>
.5 * V1 ^ 2 - G * MS / D1 = .5 * V2 ^ 2 - G * MS * D2
<=>
...bunch of algebra...
<=>
V2 = (V1 ^ 2 + .5 * G * MS (D2 ^ -1 - D1 ^ -1)) ^ .5

which leaves me with 3.36E4. Still in the right ballpark, but not quite there.

 

[davieddy]

V2 = (V1 ^ 2 + .5 * G * MS (D2 ^ -1 - D1 ^ -1)) ^ .5

which leaves me with 3.36E4. Still in the right ballpark, but not quite there.

Try
V2 = (V1 ^ 2 + 2 * G * MS (D2 ^ -1 - D1 ^ -1)) ^ .5

 

[marekkpie]

Try
V2 = (V1 ^ 2 + 2 * G * MS (D2 ^ -1 - D1 ^ -1)) ^ .5

D'oh! Tyvm.