I want to have a function that describes where a falling object is. Like this one: h(t) = -g*t²/2 But this one is for the usual close to the surface case, where there is no variation of gravity due to distance to the center of the planet. But I want a formula that takes that variation into account, for distances close to several radius. It is hypothetical situation where the only bodies involved are the the falling one and the big one. And I'm talking only about Newton physics here. I've searched for it but found nothing.
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$\begingroup$ Unless I'm mistaken, if you lift the "close to the surface" approximation, you simply get an elliptic trajectory, following Kepler's first law. $\endgroup$– MiyaseCommented Jun 26, 2022 at 22:18
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$\begingroup$ If the object has no initial velocity it has no reason to form an elliptic trajectory, that would happen if it was in orbital velocity, and it could even happen close to the surface, if it is fast enough for that and if there is no atmosphere. $\endgroup$– Ramon GriffoCommented Jun 26, 2022 at 22:20
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1$\begingroup$ I don't think that problem has an analytical solution, that is, there is no single formula of position as a function of time. $\endgroup$– user338734Commented Jun 26, 2022 at 22:24
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$\begingroup$ In the special case where initial velocity is zero, conservation of angular momentum (assuming gravitation is the only force) implies that the trajectory is a straight line, which a special case of elllipsis. Since the force follows the inverse square law, it's very likely that no analytical solution exists and that a numerical resolution is required. $\endgroup$– MiyaseCommented Jun 26, 2022 at 22:24
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2$\begingroup$ Constant-gravity parabolae are secretly high-eccentricity ellipses. The problem of motion in a $1/r^2$ gravitational field is solved, though the gravitational three-body problem is chaotic. $\endgroup$– rob ♦Commented Jun 26, 2022 at 22:49
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1 Answer
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You can solve the differential equation with varying $g$, analytically in mathematica/python/wolframAlpha to get exact time!
But if you want the time of Falling, I guess the two body problem will work! The time needed for two bodies to meet under the influence of gravitational attractive force!
$T_{meet}=\frac{π}{2\sqrt{2}}\sqrt{\frac{(R_{e}+h)^{3}-R_{e}^{3}}{G(M_{e}+m)}}$
