A couple of astronauts agree to rendezvous in space after hours. Their plan is to let gravity bring them together. She has a mass of 64.0kg and he a mass of 76.0kg , and they start from rest 23.0m apart. Find his initial acceleration. Find her initial acceleration. If the astronauts' acceleration remained constant, how many days would they have to wait before reaching each other? (Careful! They both have acceleration toward each other.) Would their acceleration, in fact, remain constant?

Question

anonymous · Answer

Woops, let me retype that. Accidentally clicked a link out of this site while typing a response. XD

anonymous · Answer

Gah! Again OpenStudy's random link popups sabotage my answer. Let me try again. >.>

anonymous · Answer

My solution... involved double integration. I don't think I'm doing it the most elegant way.

anonymous · Answer

Let's use Newton's Law of Universal Gravitation, which is expressed as$$F = G {m_1 m_2 \over r^2}$$We can find the force of attraction between them. Then, to find the acceleration from$$F = m_1 a_1 ~~ {\rm and}~~ F = m_2 a_2$$I'll take it you know the equations of motion for constant acceleration. $$y = {1 \over 2} a t^2$$

Obviously, as they accelerate towards each other, the radius between them decreases, meaning the force of gravitation between them increases, meaning they accelerate faster.

anonymous · Answer

I don't know if double integration is the best approach here. The assumption that they accelerate at a constant rate says to me that this course is being taught before differential equations.

anonymous · Answer

Where is it assumed that they accelerate at a constant rate? I know it is asked later, but it's not specified anywhere.

anonymous · Answer

Oh, wait, it is specified. Nevermind me. XD

anonymous · Answer

haha. no worries.

anonymous · Answer

Yeah, I was assuming that they weren't accelerating at a constant rate, so I was going for the position functions with a variable acceleration... which meant double integrals.

jamesj · Answer

(Actually solving this problem for non-constant acceleration (or equivalently force) is quite difficult, and involves more than double integrals; it involves solving a very non-linear differential equation.)

anonymous · Answer

You're right, I just tried to do it out and... ouch.

jamesj · Answer

[ If you're interested, have a look at this: http://openstudy.com/study#/updates/4f104a99e4b04f0f8a91fd36 ]