High School Feynman's lectures: Newton’s Laws of Dynamics

[YanaFFF]

TL;DR

Please help me figure out equations 9.13, 9.14, 9.15 from the Feynman lectures on physics (Volume 1, Chapter 9). I don't really understand what exactly these functions mean and also why they need to be added or subtracted. (Explain as simply as possible). I will be very grateful for your help!

https://www.feynmanlectures.caltech.edu/I_09.html

 

[DrClaude]

It is hard to explain things better than Feynman

For eq. (9.13), if the velocity was constant, then
$$
x(t+\epsilon) = x(t) + \epsilon v_x
$$
would be exact, as the displacement in $x$ is time elapsed ($\epsilon$) multiplied by velocity ($v_x$). Since velocity depends on time, it is only an approximation; the smaller $\epsilon$, the better.

Eq. (9.14), is the same, but for velocity in terms of acceleration. Eq. (9.15) follows from the fact that in this case acceleration is $-x$, see eq. (9.12).

 

[YanaFFF]

It is hard to explain things better than Feynman

For eq. (9.13), if the velocity was constant, then
$$
x(t+\epsilon) = x(t) + \epsilon v_x
$$
would be exact, as the displacement in $x$ is time elapsed ($\epsilon$) multiplied by velocity ($v_x$). Since velocity depends on time, it is only an approximation; the smaller $\epsilon$, the better.

Eq. (9.14), is the same, but for velocity in terms of acceleration. Eq. (9.15) follows from the fact that in this case acceleration is $-x$, see eq. (9.12).

Thank you very much! I agree that Feynman explains it well, but I still have gaps in my knowledge.