- #1
Xenn
- 6
- 0
Hello, physicsforums. I'm trying to write a proof for a function involving Newton's law of gravitation, and I seem to be stuck. The function I'm trying to build is a function of time with respect to distance.
This is the formula I want to transform.
[itex]\mathrm{A}=-\frac{GM}{x^{2}}[/itex]
For Acceleration, I expanded it to
[itex]\mathrm{A}=\frac{\mathrm{d}v}{\mathrm{d}x}\frac{\mathrm{d}x}{\mathrm{d }t}=\mathrm{V}\frac{\mathrm{d}v}{\mathrm{d}x}[/itex]
and so the equation becomes
[itex]\mathrm{V}\mathrm{d}v=-\frac{GM}{x^{2}}\mathrm{d}x[/itex]
To which I integrate and simplify to
[itex]\mathrm{V}=\sqrt{\frac{2GM}{x}}[/itex]
But V can be rewritten as dx/dt, so I've rewritten the problem to
[itex]\mathrm{d}t=\sqrt{\frac{x}{2GM}}\mathrm{d}x[/itex]
and integrate again to find time T.
Now, this is where I think I have my problem, but I've checked my math and can't find anything wrong.
It integrates and simplifies to
[itex]\mathrm{T}=\frac{\sqrt{2}x^{\frac{3}{2}}}{3\sqrt{GM}}[/itex]
However, when I plug the formula in with M = mass of the sun (1.989*10^30 kg) and x = distance between Earth and the sun( 1.496*10^11 meters ), the formula tells me it would take about 27 days to fall into the sun, when I know it would really take about 64 days.
I looked for someone who asked the same question at http://www.physlink.com/Education/AskExperts/ae226.cfm but do not understand how they got a different formula. I am trying to also stay away from Kepler's Laws, so this should be a pure integration/transformation problem.
This is the formula I want to transform.
[itex]\mathrm{A}=-\frac{GM}{x^{2}}[/itex]
For Acceleration, I expanded it to
[itex]\mathrm{A}=\frac{\mathrm{d}v}{\mathrm{d}x}\frac{\mathrm{d}x}{\mathrm{d }t}=\mathrm{V}\frac{\mathrm{d}v}{\mathrm{d}x}[/itex]
and so the equation becomes
[itex]\mathrm{V}\mathrm{d}v=-\frac{GM}{x^{2}}\mathrm{d}x[/itex]
To which I integrate and simplify to
[itex]\mathrm{V}=\sqrt{\frac{2GM}{x}}[/itex]
But V can be rewritten as dx/dt, so I've rewritten the problem to
[itex]\mathrm{d}t=\sqrt{\frac{x}{2GM}}\mathrm{d}x[/itex]
and integrate again to find time T.
Now, this is where I think I have my problem, but I've checked my math and can't find anything wrong.
It integrates and simplifies to
[itex]\mathrm{T}=\frac{\sqrt{2}x^{\frac{3}{2}}}{3\sqrt{GM}}[/itex]
However, when I plug the formula in with M = mass of the sun (1.989*10^30 kg) and x = distance between Earth and the sun( 1.496*10^11 meters ), the formula tells me it would take about 27 days to fall into the sun, when I know it would really take about 64 days.
I looked for someone who asked the same question at http://www.physlink.com/Education/AskExperts/ae226.cfm but do not understand how they got a different formula. I am trying to also stay away from Kepler's Laws, so this should be a pure integration/transformation problem.