A mass hanging from a spring is set in motion and its ensuing velocity is given by \( v(t)=2 \pi \cos \pi t \) for \( t \geq 0 \). Assume that the positive direction is upward and \( s(0)=0 \). a. Determine the position function for \( t \geq 0 \). b. Graph the position function on the interval \( [0,3] \) c. At what times does the mass reach its lowest point the first three times? d. At what times does the mass reach its highest point the first three times?

To solve the problem, we start with the provided velocity function: ### a. Determine the position function \( s(t) \). The velocity function is given by \( v(t) = 2\pi \cos(\pi t) \). The position function \( s(t) \) can be found by integrating the velocity: \[ s(t) = \int v(t) \, dt = \int 2\pi \cos(\pi t) \, dt \] Using the integral of cosine, we get: \[ s(t) = \frac{2\pi}{\pi} \sin(\pi t) + C \] This simplifies to: \[ s(t) = 2 \sin(\pi t) + C \] To find the constant \( C \), we use the initial condition \( s(0) = 0 \): \[ s(0) = 2 \sin(0) + C = 0 \implies C = 0 \] Thus, the position function is: \[ s(t) = 2 \sin(\pi t) \] ### b. Graph the position function on the interval \( [0, 3] \). The graph of \( s(t) = 2 \sin(\pi t) \) for \( t \) in the interval \( [0, 3] \) oscillates between -2 and 2. The sine function has a period of 2, so it completes one full cycle (from 0 to its maximum at \( t = 0.5 \), back to 0 at \( t = 1 \), down to its minimum at \( t = 1.5 \), and returning to 0 at \( t = 2 \)) within that interval. You would see two full oscillations in the range up to \( t = 3 \). ### c. At what times does the mass reach its lowest point the first three times? The lowest point occurs when \( s(t) \) is at a minimum. The sine function \( \sin(\pi t) \) reaches -1 at: \[ \pi t = \frac{3\pi}{2} + 2k\pi \quad \text{(for integer } k\text{)} \] \[ t = \frac{3}{2} + 2k \quad \text{(for integer } k\text{)} \] Calculating the first three instances for \( k = 0, 1, 2 \): - For \( k = 0: t = 1.5 \) - For \( k = 1: t = 3.5 \) - For \( k = 2: t = 5.5 \) Thus, the mass reaches its lowest point the first three times at approximately \( t = 1.5, 3.5, 5.5 \). ### d. At what times does the mass reach its highest point the first three times? The highest point occurs when \( s(t) \) is at a maximum. The sine function \( \sin(\pi t) \) reaches 1 at: \[ \pi t = \frac{\pi}{2} + 2k\pi \quad \text{(for integer } k\text{)} \] \[ t = \frac{1}{2} + 2k \quad \text{(for integer } k\text{)} \] Calculating the first three instances for \( k = 0, 1, 2 \): - For \( k = 0: t = 0.5 \) - For \( k = 1: t = 2.5 \) - For \( k = 2: t = 4.5 \) Thus, the mass reaches its highest point the first three times at approximately \( t = 0.5, 2.5, 4.5 \).