# X-t plot of underdamped motion

Here is the x-t plot of underdamped motion:

At the intersection of the red curve and the blue curve, the red curve has a slope of 0 but the blue curve has a nonzero slope.

Therefore, does it mean that these two curves actually intersect at some other points very close to the intersections already show in the plot?

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AlephZero
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Don't try to read too much into the plot. It's not very accurately drawn. You can see some "kinks" in the top blue curve at about t= 0.2 and t= 0.4 which shouldn't be there.

Most likely it's a plot of $e^{-at}$ and $e^{-at}\cos(bt)$ (you can guess the values of a and b for yourself).

EDIT: first comment was wrong!

When $\cos(bt) = 1$, the slope of both curves is $-a e^{-at}$ so the two curves touch at one point.

But the curves in your link are probably drawn by fitting a "smooth curve" through a set of points, and so they will not necessarily have the correct tangents and the two curves look as if they intersect twice.

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But mathematically there shouldn't be any reason that the equation exp(-at)=exp(-at)cos(bt) have 2 solutions at each peak.

Or it might be the case that the red curve does not have a zero slope at the intersection. (Because exp(-at)cos(bt) and cos(bt) reach max/min at different t)

AlephZero