Potential energy turned into kinetic

[cragar]

Homework Statement

We have an object with much smaller mass than compared to the sun.
This object is at rest when released at 1AU from the sun. After a week how far is the mass from the sun.

The Attempt at a Solution

So I look at the gravitational potential energy and set it equal to kinetic energy.
$\frac{-GMm}{r}=\frac{m(v)^2}{2}$
the v should be an r dot. now I take the square root of both sides. and then multiply both sides by dt then move the r from the left side to the right side and then integrate bothe sides. this will give me r(t). and then for the bounds I use r=1au and then I can solve for r final. after using t=0 and t=1 week for the time. i will put the time in seconds. this seems somewhat reasonable.

 

[Orodruin]

Why would the kinetic energy equal the gravitational potential? There is nothing saying this has to be the case (furthermore, the potential is only defined up to an arbitrary constant) and in particular it does not make sense in your case since the potential is strictly negative and kinetic energy non negative.

You can go one of two ways:
1. Write down the force equation and solve for r as a function of time.
2. Use conservation of energy (this will save you one integration)

 

[cragar]

for the force equation would I use F=ma and set it equal to the gravitational force.
For conservation of energy would I say that kinetic plus potential is a constant.

 

[Orodruin]

Yes, both would be correct approaches so I suggest you try doing either of those and return with the result. For the second approach, remember that $v = -\dot r$ and (for both approaches) that $v(t=0) = 0$ and $r(t=0) = R_\oplus$.