I have read this article https://physics.stackexchange.com/questions/3534/dont-heavier-objects-actually-fall-faster-because-they-exert-their-own-gravity.
In the best answer by David Z - there are some mathematical tricks I don't understand.
First there is the equation: $$\frac{G(m_1 + m_2)}{r^2} = -\ddot{r}$$
Then:
So it's clear that when the total mass is larger, the magnitude of the acceleration is larger, meaning that it will take less time for the objects to come together. If you want to see this mathematically, multiply both sides of the equation by r˙dt to get
$$\frac{G(m_1 + m_2)}{r^2}\mathrm{d}r = -\dot{r}\mathrm{d}\dot{r}$$
Up to this point I understand the mathematics but here is the problem:
According to the answer, by integration I should get $$G(m_1 + m_2)\left(\frac{1}{r} - \frac{1}{r_i}\right) = \frac{\dot{r}^2 - \dot{r}_i^2}{2}$$
Question 1 What has been done now? $G(m_1 + m_2)$ is a constant so it stays the same. But: $$\int\frac{1}{r^2}dr= -\frac{1}{r}$$ Why is there the $\left(\frac{1}{r} - \frac{1}{r_i}\right)$??
Question 2 How write the right side of this equation $$\frac{G(m_1 + m_2)}{r^2}\mathrm{d}r = -\dot{r}\mathrm{d}\dot{r}$$ in a way like this: $$-\dot{r}= -1 \cdot\dfrac{dr}{dt}$$ I don't know what to do with $\mathrm{d}\dot{r}$.
Question 3 Why is $$\int -\dot{r}\mathrm{d}\dot{r}= \frac{\dot{r}^2 - \dot{r}_i^2}{2} $$
I don't really get the steps that are not written there. To me it seems that the equation could also be written as $$\int -x \,\mathrm{d}x= -0.5 x^2$$ Now the answers are not the same.
Question 4
$$G(m_1 + m_2)\left(\frac{1}{r} - \frac{1}{r_i}\right) = \frac{\dot{r}^2 - \dot{r}_i^2}{2}$$ Assuming $\dot{r}_i = 0$ (the objects start from relative rest), you can rearrange this to $$\sqrt{2G(m_1 + m_2)}\ \mathrm{d}t = -\sqrt{\frac{r_i r}{r_i - r}}\mathrm{d}r$$
What has been done right know?
How is $$\dot{r}_i = 0$$ related to $$- \frac{1}{r_i}$$?
I don't get this at all.
I know that the derivation goes on but in my opinion it doesn't make sense trying to understand the rest if I haven't got it until here. My mathematics knowledge is that of the European equivalent of "academic high school".
It would be really cool to get some helpful answers.
I have already started with: http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/index.htm but I haven't come that far yet.
Nevertheless I am willing to understand this derivation.
Question 5 What is the best way to learn such things like my question. Shall I continue with OCW or shall I do something else?