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here is the topic of the problem:

You are given $2$ baseballs (consider them as perfect solid spheres) have equal properties with mass $m = 0,142kg$, radius $r_0 = 0.037m$ in the space and thay are $1m$ apart (the distance between their centres of mass), both of their initial velocities are $0$, calculate how long they will collide to (touch) each other? Given gravitational constant is $G = 6,67408 \cdot 10^{-11}$.

Here is my solution:

Since the system is symmetric, we just analyse $1$ ball, let's consider it is ball $m_1$.

Let's call the distance function between those $2$ balls $m_1$ and $m_2$ is $r(t) (m)$ which is depended on time $t$, in which $r(0) = 1 (m)$. (*)

Therefore the question we are looking for can be understanded as in which time $t$ that can make the function $r(t) = 2 \cdot r_0= 0.074 (m)$ (find $t$)

We got gravitational force equation changes through time is: $F(t) = G \cdot \frac{m^2}{r^2(t)} (N)$ (notice that $m_1 = m_2 = m)$

Acceleration function for ball $m_1$ is: $a(t) = \frac{F(t)}{m_1} = G \cdot \frac{m}{r^2(t)} \approx \frac{9.477 \cdot 10^{-12}}{r^2(t)} (m/s^2)$

We notice that $a(t) = r''(t) = \frac{9.477 \cdot 10^{-12}}{r^2(t)}$

$\Leftrightarrow r''(t) - \frac{9.477 \cdot 10^{-12}}{r^2(t)} = 0$ (**)

I think from (*) and (**) we can solve for the general form of this equation. However, I'm still a secondary school student right now, so at this step, my mathematical knowledge is not ripe enough for these kinds of equation, as I did some researches I knew that these kinds of equations are called "second-order differential equations" or something like that, and literally I have no idea how to deal with them. Can someone help me through this problem please? I'm dedicatedly seeking for the answer, thank you!

by the way, this is not homework exercise or something, I'm just a math, physics enthusiast and I'm kinda bored right now so I come up with this problem myself

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  • $\begingroup$ Hint: If you multiply by $r’(t)$, both terms will become the derivative of something. $\endgroup$
    – Ghoster
    Commented Apr 22, 2023 at 4:17
  • $\begingroup$ you can consider this is some kind of "homework asking" question, but to be honest, I have no idea how to solve differential equation, can you just give me the answer please ? $\endgroup$
    – Michael
    Commented Apr 22, 2023 at 4:19
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    $\begingroup$ Can you just give me the answer please? No, sorry, it would violate the policies of this site to provide a complete answer to a homework-like question like this. But I’m confused… if you don’t know anything about differential equations, why have you been assigned a problem that involves a differential equation? $\endgroup$
    – Ghoster
    Commented Apr 22, 2023 at 4:39
  • $\begingroup$ Maybe you would understand how to solve it using the conservation of energy instead of Newton’s Second Law. That still involves a differential equation, but one that involves only first-order derivatives. (It’s equivalent to the equation you’d get by following my hint.) $\endgroup$
    – Ghoster
    Commented Apr 22, 2023 at 4:45
  • $\begingroup$ Since this is a standard problem (often posed in terms of the Earth falling from rest into the Sun), you can certainly find solutions on the web, and probably in other questions on this site. $\endgroup$
    – Ghoster
    Commented Apr 22, 2023 at 4:49

1 Answer 1

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You might solve this without any advanced math in excel or other spreadsheet program, as follows:

enter image description here

At separation distance lower than 0.074m, the balls will have touched, so we might eventually read off the time value for the two balls to touch (if the spreadsheet's row limit will allow for that many rows).

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  • $\begingroup$ You haven’t shown the Excel formulas for the velocity or position. Are you using the acceleration calculated in B4? If so, that’s only the initial acceleration. $\endgroup$
    – Ghoster
    Commented Apr 22, 2023 at 5:20
  • $\begingroup$ @ghoster oh yes, the acceleration must be recalculated every step. i will update. $\endgroup$
    – James
    Commented Apr 22, 2023 at 5:23
  • $\begingroup$ The balls separated by 1 meter will hit each other in 2.93 days. The earth and the sun in the same situation will hit in 65.05 days. $\endgroup$
    – James
    Commented Apr 22, 2023 at 11:41
  • $\begingroup$ @James hi, can I have your excel file please ? I'm not very good at excel $\endgroup$
    – Michael
    Commented Apr 22, 2023 at 14:33
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    $\begingroup$ @James 2.93 days This agrees with the analytic formula. $\endgroup$
    – Ghoster
    Commented Apr 22, 2023 at 16:05

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