- #1
Crush1986
- 207
- 10
Homework Statement
An asteroid is knocked out of the Kuiper belt and starts to fall toward the sun. (Assume it's initial potential energy and kinetic energy is 0.)
a. Write down the differential equation for r(t) giving the position of the asteroid as a function of time.
b. Solve the differential equation for t(r), the time as a function of position
Homework Equations
[tex] F=\frac{GmM}{r^2} [/tex]
The Attempt at a Solution
I wrote out what I think the differential equation is from the Newtons second law diagram.
[tex] \frac{dr^2}{dt^2}=\frac{GM}{r^2} [/tex]
I think I can rewrite to second derivative of r with respect to time as velocity multiplied by the derivative of velocity with respect to r. Making my equation
[tex] v\frac{dv}{dr}=\frac{GM}{r^2} [/tex]
I integrate this to get
[tex] \frac{v^2}{2}=\frac{-GM}{r} [/tex]
I stopped here because I'm not sure if maybe I should do this integration with some limits? Like from R (my starting point) to r (current position). What should I make the limits for velocity? 0 to v?
More importantly, I'm very very new to differential equations. Is all this even remotely correct? I imagine that if I'm correct up to this point I can just solve for v and take one more derivative and I think I'll be very close to finished.
Any help is greatly appreciated!