Help with Gravity question

Question

I need some help with this question from my homework.

An Oort cloud comet moves on an orbit so eccentric that it can be well approximated by a radial straight line through the Sun. The comet starts at a large distance from the Sun, with zero velocity, and falls inward. (a) What is the comet’s speed when it reaches a distance of 1 AU from the Sun?

So I am lost on really how to start. I can't use the force of gravity because we aren't given the mass of the comet or the distance between that and the sun. I can't use $v_f^2 = v_i^2 + 2ad$ because we don't have a constant acceleration.

Maybe this is supposed to be a problem where I use variables for the mass and distance? But even then, the distance changes as we approach the sun, which changing the force. I guess I'd need to integrate something? I really have no clue how to start, and any help would be appreciated.

Answer 1

Have you learned about gravitational potential? The force of gravity follows an inverse square law, $F=\frac{GM_1M_2}{r^2}$; the potential, which is the integral $\int F dr$, goes as $-\frac{GM_1M_2}{r}$. As you go from "infinity" (where the potential is zero) to some smaller radius, some potential energy is released and converted to kinetic energy, $\frac12 m v^2$.

The mass of the comet cancels out. Take it from there...