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## Homework Statement

I have this problem where the earth immediately loses all orbital velocity and begins to fall towards the sun, and I need to find the time it takes for the earth to hit it.

Seemed straight forward enough.

## Homework Equations

Started with the work k.e. theorem,

.5mv(x)

^{2}-.5mv

_{o}

^{2}=∫[from x

_{o}to x]F(x)dx.

Where m=the mass of the earth, v

_{o}=0, and F(x)=F(r)=-GMm/r

^{2}where M is the mass of the sun.

## The Attempt at a Solution

So I made the bounds translate from x

_{o}→x to r

_{AU}→r(t). Solved the integral and got

v(x)=sqrt(-2GM((1/r

_{AU})-(1/r(t))))

Seeing as how r(t) is never going to get bigger than 1 AU, this doesn't make any sense. The answer is already imaginary, and I haven't even gotten to the integral for solving for t(r) yet. Anybody know what I did wrong?

note: In case it wasn't that obvious, I'm using the initial position of the Earth as r

_{AU}and the final point I'm trying to get the Earth to is the radius of the sun, ill just write as r

_{o}. The center of the sun is the origin of the coordinate system.