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## Homework Statement

For a smooth (“low jerk”) ride, an elevator is programmed to start from rest and accelerate according to

$$a(t) = \frac{a_m}{2}[1 − \cos{\frac{2\pi t}{T}}] \:\:\:\:0 ≤ t ≤ T$$

$$a(t) = -\frac{a_m}{2}[1 − \cos{\frac{2\pi t}{T}}] \:\:\:\:T ≤ t ≤ 2T$$

Where ##a_m## is the maximum acceleration and ##2T## is the total time for the trip.

(a) Draw sketches of ##a(t)## and the jerk as functions of time.

(b) What is the elevator’s maximum speed?

(c) Find an approximate expression for the speed at short times near the start of the ride, ##t ≪ T##.

2. Homework Equations

2. Homework Equations

##j = \dot{a} = \ddot{v}##

## The Attempt at a Solution

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The sketch required for part (a) is trivial, so I won't go over the details of the two sketches.

Now, for part (b), I found the value of ##t## for which the acceleration is zero, and used integration to find the velocity (or rather, maximum velocity) at that value of ##t##. The problem is, there's more than one value of ##t## for which the acceleration of the elevator is zero. In fact, the acceleration is zero at ##t = 0## inspite of the fact that ##v(0) = 0##. I feel like I have made some kind of error somewhere.