Question

examine the following graph of a function modeling damped harmonic motion. find the equation for the function pictured in terms of y and t. assume that a factor of e^(-t) provides the desired damping effect and that the graph has no vertical or horizontal shifts.

Explanation:

Step1: Identify the general form

The general form of a damped - harmonic motion function with no vertical or horizontal shifts is $y = Ae^{-bt}\sin(ct)$ or $y = Ae^{-bt}\cos(ct)$. Since the graph starts at $y = 4$ when $t = 0$, we use the cosine - based form $y=Ae^{-bt}\cos(ct)$. Given that $b = 1$ (from the damping factor $e^{-t}$), and when $t = 0$, $y=Ae^{0}\cos(0)=A$. From the graph, when $t = 0$, $y = 4$, so $A = 4$.

Step2: Find the value of $c$

The period of the undamped cosine function (ignoring the damping for now) can be estimated from the graph. The distance between two consecutive peaks gives an estimate of the period. From the graph, the period $T$ of the undamped - like motion is approximately $1$. The formula for the period of a cosine function $y=\cos(ct)$ is $T=\frac{2\pi}{c}$. Since $T = 1$, we have $1=\frac{2\pi}{c}$, so $c = 2\pi$.

Answer:

$y = 4e^{-t}\cos(2\pi t)$