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I'm having some trouble with this problem (adapted from K&K 1.23).
An elevator is programmed to start from rest and accelerate according to
\begin{align*}
a(t) &= \frac{a_m}{2}\left[1 - \cos\left(\frac{2\pi t}{T}\right)\right] &\mbox{for }0\leq t\leq T \\
a(t) &= -\frac{a_m}{2}\left[1 - \cos\left(\frac{2\pi t}{T}\right)\right] &\mbox{for }T\leq t\leq 2T
\end{align*} where ##a_m## is the maximum acceleration and ##2T## is the total time for the trip.
What is the time required for a trip of distance ##D##?
##v = \int adt##
##x = \int vdt##
Assume for a moment that ##0 \leq t \leq T##. Integrating then yields
\begin{align*}
x &= \frac{a_m}{2}\left[\frac{t^2}{2} + \left(\frac{T}{2\pi}\right)^2\cos\frac{2\pi t}{T}\right]
\end{align*}
Letting ##t = 0##, we find
\begin{align*}
x_0 &= \frac{a_m}{2}\left(\frac{T}{2\pi}\right)^2
\end{align*}
It follows that
\begin{align*}
D &= x - x_0 = \frac{a_m}{2}\left[\frac{t^2}{2} + \left(\frac{T}{2\pi}\right)^2\left(\cos\frac{2\pi t}{T} - 1\right)\right]
\end{align*}
But I'm not seeing a way to hammer this into a useful expression for ##t##. Perhaps there is an error in the way the problem is worded?
Homework Statement
An elevator is programmed to start from rest and accelerate according to
\begin{align*}
a(t) &= \frac{a_m}{2}\left[1 - \cos\left(\frac{2\pi t}{T}\right)\right] &\mbox{for }0\leq t\leq T \\
a(t) &= -\frac{a_m}{2}\left[1 - \cos\left(\frac{2\pi t}{T}\right)\right] &\mbox{for }T\leq t\leq 2T
\end{align*} where ##a_m## is the maximum acceleration and ##2T## is the total time for the trip.
What is the time required for a trip of distance ##D##?
Homework Equations
##v = \int adt##
##x = \int vdt##
The Attempt at a Solution
Assume for a moment that ##0 \leq t \leq T##. Integrating then yields
\begin{align*}
x &= \frac{a_m}{2}\left[\frac{t^2}{2} + \left(\frac{T}{2\pi}\right)^2\cos\frac{2\pi t}{T}\right]
\end{align*}
Letting ##t = 0##, we find
\begin{align*}
x_0 &= \frac{a_m}{2}\left(\frac{T}{2\pi}\right)^2
\end{align*}
It follows that
\begin{align*}
D &= x - x_0 = \frac{a_m}{2}\left[\frac{t^2}{2} + \left(\frac{T}{2\pi}\right)^2\left(\cos\frac{2\pi t}{T} - 1\right)\right]
\end{align*}
But I'm not seeing a way to hammer this into a useful expression for ##t##. Perhaps there is an error in the way the problem is worded?