Problem: A girl moves her head back and forth in time to the beat of some rock music:

Assuming that the muscular control obeys the above formula (so that the motion is SHM),

a) How far does her head move?
b) Taking to her right as positive, where does her head start? With what velocity?
c) What are the period, frequency, and angular frequency of the motion?
d) What are the maximum speed and acceleration of her head?
e) What are the velocity and acceleration at t = 1.5 seconds?
f) When her head is first 12 cm from the central position (x = 0), what is the velocity?

a) Her head moves from +18 cm to -18 cm, for a total of 36 cm.

b) At t = 0, x = 0.09 m, so her head starts at + 9 cm. Taking a derivative,

At t = 0, the velocity is -0.490 m/s, so she doesn't start with her head at its maximum position, and her head has an initial velocity toward the center.

Since a "cycle" is a counting number & not a unit, it can be dropped.

The maximum speed is 0.565 m/s (no sign, since a scalar was requested). The maximum acceleration (magnitude) is 1.78 m/s².

f) When is her head first 12 cm from x = 0? Substituting into the equation for position...

You'll notice that 1) the calculator has to be set for radians, and 2) we have to consider a series of possible answers (there's an infinite number, but three should be enough). Solving these for t, we get

I've shown the calculation for just the first angle in each case. You can see that the first line results in a negative time, so that one is discarded. The second line (0.398 s) is a possible time, corresponding to x = -12 cm, but is it the earliest time? (Thinking about the physical motion, her head moves from +9 cm to zero and then to -12 cm, so this is a sensible answer.)

You should verify that you can obtain these: Using the other values, I got 1.40 s from the 2Pi-0.841 angle, and 0.935 s from the 2Pi-2.30 angle. So the 0.398 s is the earliest time.

Substituting the 0.398 s into the velocity formula, I get -0.422 m/s.