Passing the time I decided to watch some lectures and it was given that the electron in a neutral hydrogen atom would impact the proton on the order of 0.1 ns in the most rudimentary model. While the math was not the focus of the lecture I wanted to try this myself but get an answer on the order of $10^{-76}$ for an initial position of $5\times10^{-11} \text{m}$.
I used $F = m\frac{d^2r}{dt^2}$ and $F_C = \frac{q_1 q_2}{4 \pi \epsilon_0}\frac{1}{r^2}$ and related the quantities as
$$dt^2 = \frac{m}{F_C} d^2r$$
then tried to solve by integrating both sides twice.
This did not give the correct answer. Whereas this approach has worked for first order derivatives (if memory serves, it's been many years), it doesn't for the second order. I spent a lot of time looking for a way forward and came across a post emphasizing that "derivatives are not fractions." This was the indicator that I wasn't just a little off but way off.
How do I approach this problem or can you give me a topic I need to review to be able to solve it?
Thank you!
