X-t plot of underdamped motion

[AlonsoMcLaren]

Here is the x-t plot of underdamped motion:

http://www.google.com/imgres?q=unde...2&tbnw=151&start=0&ndsp=20&ved=1t:429,r:2,s:0

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?

 

[AlephZero]

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.

 

[AlonsoMcLaren]

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

 

[AlonsoMcLaren]

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]

Sorry, my first comments in post #2 were wrong - now fixed.