B Feynman's lectures: Newton’s Laws of Dynamics

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Feynman's lectures on Newton's Laws of Dynamics emphasize the relationship between displacement, velocity, and time. In equation (9.13), displacement is approximated as the product of time elapsed and velocity, which becomes more accurate with smaller time increments. Equation (9.14) relates velocity to acceleration, while equation (9.15) highlights that acceleration equals negative displacement. Despite Feynman's clarity, some readers still feel gaps in their understanding of these concepts. The discussion underscores the effectiveness of Feynman's teaching style in conveying complex physics principles.
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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!
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It is hard to explain things better than Feynman :smile:

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).
 
DrClaude said:
It is hard to explain things better than Feynman :smile:

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.
 
This has been discussed many times on PF, and will likely come up again, so the video might come handy. Previous threads: https://www.physicsforums.com/threads/is-a-treadmill-incline-just-a-marketing-gimmick.937725/ https://www.physicsforums.com/threads/work-done-running-on-an-inclined-treadmill.927825/ https://www.physicsforums.com/threads/how-do-we-calculate-the-energy-we-used-to-do-something.1052162/
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