91Ó°ÊÓ

\(s(t) = 2\sin(\pi t) + C\) To determine the constant of integration C, we will use the initial condition: when t = 0, the position is 2 cm above the equilibrium (s(0) = 2). \(s(0) = 2\sin(\pi \cdot 0) + C = 2\) \(C = 2\) Therefore, the position function is: \(s(t) = 2\sin(\pi t) + 2\) for \(t ≥ 0\) #b. Graph the position function on the interval [0, 2]# Using graphing software or a graphing calculator, graph the position function \(s(t) = 2\sin(\pi t) + 2\) on the interval [0, 2]. The graph should display a sinusoidal wave with an amplitude of 2 and a period of 2. #c. Determine the times when the mass reaches its low and high points for the first three times# To find the high points, we identify the points where the derivative of the position function, which is the velocity function, changes from positive to negative. The velocity function is \(v(t) = 2\pi \cos(\pi t)\). Set it equal to zero and solve for t: \(2\pi \cos(\pi t) = 0\) \(\cos(\pi t) = 0\) Since a cosine function equals zero at odd multiples of \(\frac{\pi}{2}\), the high points occur when: \(\pi t = \frac{\pi}{2} + 2k\pi\), where \(k\) is any integer. Now, solve for the three smallest \(t\) values: 1) \(k = 0: \pi t = \frac{\pi}{2} \Rightarrow t = \frac{1}{2}\) 2) \(k = 1: \pi t = \frac{\pi}{2} + 2\pi \Rightarrow t = \frac{5}{2}\) 3) \(k = 2: \pi t = \frac{\pi}{2} + 4\pi \Rightarrow t = \frac{9}{2}\) The high points occur at \(t = \frac{1}{2}\), \(\frac{5}{2}\), and \(\frac{9}{2}\). Similarly, to find the low points, we identify points where the velocity changes from negative to positive. Since a cosine function equals zero at odd multiples of \(\pi\), the low points occur when: \(\pi t = \pi + 2k\pi\), where \(k\) is any integer. Again, solve for the three smallest t values: 1) \(k = 0: \pi t = \pi \Rightarrow t = 1\) 2) \(k = 1: \pi t = 3\pi \Rightarrow t = 3\) 3) \(k = 2: \pi t = 5\pi \Rightarrow t = 5\) The low points occur at \(t = 1\), \(3\), and \(5\). In summary, the position function for the mass hanging from the spring is \(s(t) = 2\sin(\pi t) + 2\) for \(t ≥ 0\). The high points are at \(t = \frac{1}{2}\), \(\frac{5}{2}\), and \(\frac{9}{2}\) while the low points are at \(t = 1\), \(3\), and \(5\).