Sinusoidal function

Question

The propeller of a boat at dock in the ocean will rise and fall with the waves. On a particularly wavy night, the propeller leaves its resting position and reaches a height of 2m on the peaks of the waves and -2m in the troughs. The time between the peak and the trough is approximately 3 seconds. Determine the equation of a sinusoidal function that would model this situation assuming that at t=0 the propeller is at its resting position and headed towards the peak of the next wave.

Answer 1

Julia,

 

Based upon what you have told me in your question, the best function to model the height of the propeller is a sine function. You already know the amplitude is 2 m (meters?), and the period is 6 sec (twice the time from the high to low tide).

 

Use the equation y = A sin (Bt). With A = 2 and 2π/B = 6, you can solve for B using Algebra:

 

2π = 6B  ⇒  B = π/3.

 

Plug this into the equation, you get

 

y = 2 sin (π t/3).

 

As a check, the peak will be reached after ¼ of the period, or 1.5 sec. If you plug this in for t, you get y = 2 sin (.5π) = 2 sin (π/2) = 2. After 3 more sec, or t = 4.5 sec, the height is y = 2 sin (1.5π) = 2 sin (3π/2) = −2. This agrees with what was give in the problem.

Answer 2

With a quantity that you know is described by a sinusoidal function, the two choices (at this level) tend to be sine and cosine, both of which take a similar form described by Alan in the previous answer.

 

Just to fill in a bit on why sine is the choice in this case, and not cosine:  it relates to what we are told about the initial conditions at time t=0.  Since it is at the "resting position" (displacement = 0) at t=0, this position must be described by a function that gives a result of 0 when t=0.  Between Asin(Bt) and Acos(Bt), with B≠0, this must be Asin(Bt), since when t=0, we have Asin(0), and sin(0) = 0.  Acos(0) would give a non-zero answer, since cos(0) = 1.

 

The other initial condition given is that, at t=0, it is moving toward the peak of the next wave; in other words, up.  If we define up as positive, this means that the position should be described by Asin(Bt) rather than -Asin(Bt), since the position has to be positive at short times after t=0.  If it have been heading toward a valley, we would have had to choose -Asin(Bt).  

 

A and B are defined in terms, respectively, of the amplitude and period, as Alan described, so you can fill in the values as he showed.