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If the ball is dropped downwards from a huge distance like 100 km, ignoring air resistance, and accounting for the gravitational acceleration at every point that the object is in between its starting and ending point.

Soumil
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  • How have you tried to solve the problem? What resources have you looked at? – Anders Sandberg Jan 13 '20 at 00:45
  • https://en.wikipedia.org/wiki/Free_fall#Inverse-square_law_gravitational_field – G. Smith Jan 13 '20 at 04:01
  • This problem involves solving a differential equation. Have you studied differential equations yet? The easiest approach is to write the differential equation for kinetic+potential energy conservation and integrate it to get the time. – G. Smith Jan 13 '20 at 04:07
  • 100km is not a huge distance compared with the radius of the Earth (6370km). For such a height you can assume that $g$ is constant. – sammy gerbil Jan 13 '20 at 04:10
  • @AndersSandberg, i first thought of this when I was solving a different problem that was asking for the difference in time for two objects falling at different heights, based on the difference in the gravitational acceleration at their heights. It wasn't a significant difference because the difference in heights was ~1000 meters, but I started wondering about the change in the acceleration as the object got closer to the earth, and how it would affect the time, and for a lot greater distances. – Soumil Jan 13 '20 at 04:34
  • @G.Smith, I'm currently a freshmen in high school, and I haven't taken a calculus course yet, so I don't really know anything about differentials except that it's something related to a small change in a value with respect to a change over time. – Soumil Jan 13 '20 at 04:37
  • @sammygerbil, 100 km was just an example i used, but any distance that's significant enough, 1000km, 2000, applies the same to my question – Soumil Jan 13 '20 at 04:38
  • Since the gravitational force is constantly changing as the ball falls, you have to use calculus to correctly solve problems like this. Until you learn calculus, you use the approximation where the force stays constant. Even over 100 km, this is a good approximation because 100 km is a small fraction of the Earth’s radius. But over thousands of km, it is not a good approximation. This should motivate you to learn calculus, perhaps early via self-study. – G. Smith Jan 13 '20 at 04:50
  • @Soumil Ok see linked question and the references in the linked column of that. It's a very difficult calculation, but there's a clever trick to find time for 2 point masses to collide - see Pulsar's answer in The Time That 2 Masses Will Collide Due To Newtonian Gravity.

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     – sammy gerbil Jan 13 '20 at 04:54
  • @sammygerbil, thanks, I'll check it out. – Soumil Jan 13 '20 at 05:00
  • @G.Smith, ok thanks, do you know of any good resources to learn calculus? Any books or online resources, because I won't be learning it till my junior year. – Soumil Jan 13 '20 at 05:01
  • @Soumil One possibility might be this online MIT course. – G. Smith Jan 13 '20 at 08:58

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