Suppose a ball of finite mass is taken to such a height in space that the gravitational acceleration decreases significantly. Now, as you let go of the ball, it should head straight down towards the surface of earth, and as it crosses distance and gets closer to the earth surface, it's acceleration should increase. How would I, in this case, be able to calculate the distance traveled by the ball in a certain time?
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1$\begingroup$ Do you understand what differential equations are? $\endgroup$– G. SmithCommented Feb 8, 2020 at 5:53
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$\begingroup$ en.wikipedia.org/wiki/… $\endgroup$– G. SmithCommented Feb 8, 2020 at 5:54
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1$\begingroup$ Duplicate of How to find distance travelled when the change in the force of gravity is not negligible? and probably many others. $\endgroup$– G. SmithCommented Feb 8, 2020 at 5:59
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$$\begin{align}F&=-\frac {GMm}{x^2}\\ a&=-\frac {GM}{x^2}\\ v\frac {dv}{dx}&=-\frac {GM}{x^2}\\ \int v\;{dv}&=\int-\frac {GM}{x^2}\;dx\\ \frac {v^2}2&=\frac {GM}x\\ v&=\sqrt\frac {2GM}x\\ \int\sqrt x\;dx&=\int\sqrt {2GM}\;dt\\ \frac 23x^{\frac 32}&=t\sqrt {2GM} \end{align}$$
Here, you have an equation relating distance and time in a gravitational field.
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$\begingroup$ This is valid only when the object has zero total energy, which is not the case in the OP’s question. And, with these signs, the object is rising, not falling as the OP asked. $\endgroup$– G. SmithCommented Feb 8, 2020 at 17:25
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$\begingroup$ I derived a general equation and as this is a homework question, I find it reasonable to let the OP figure out for himself how to use it. $\endgroup$– SamCommented Feb 8, 2020 at 18:28
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$\begingroup$ It is not a general equation at all. It applies to a specific case which is not the requested case. $\endgroup$– G. SmithCommented Feb 8, 2020 at 18:29
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$\begingroup$ The general equation is in the Wikipedia link. There is no simple formula for position as a function of time, only for time as a function of position. There is no indication that this is a homework problem. In fact, it usually is not given as homework because the integral is messy. Your case is the simple homework case. $\endgroup$– G. SmithCommented Feb 8, 2020 at 18:50