# Can someone help me with differential equation please? [duplicate]

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

• Hint: If you multiply by $r’(t)$, both terms will become the derivative of something. Commented Apr 22, 2023 at 4:17
• 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 ? Commented Apr 22, 2023 at 4:19
• 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? Commented Apr 22, 2023 at 4:39
• 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.) Commented Apr 22, 2023 at 4:45
• 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. Commented Apr 22, 2023 at 4:49