A mass hanging from a spring is set in motion and its ensuing velocity is given by \( v(t)=-3 \pi \sin \pi t \) for \( t \geq 0 \). Assume that the positive direction is upward and \( s(0)=3 \). 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?

Let's analyze the motion of the mass-spring system step by step. ### Given: - **Velocity Function:** \( v(t) = s'(t) = -3\pi \sin(\pi t) \) for \( t \geq 0 \) - **Initial Position:** \( s(0) = 3 \) meters (assuming meters as units) ### a. **Determine the Position Function \( s(t) \) for \( t \geq 0 \).** To find the position function \( s(t) \), we integrate the velocity function: \[ s(t) = \int v(t) \, dt + C = \int -3\pi \sin(\pi t) \, dt + C \] Let's perform the integration: \[ \begin{align*} s(t) &= -3\pi \int \sin(\pi t) \, dt + C \\ &= -3\pi \left( -\frac{\cos(\pi t)}{\pi} \right) + C \quad \text{(since } \int \sin(\pi t) \, dt = -\frac{\cos(\pi t)}{\pi} \text{)} \\ &= 3 \cos(\pi t) + C \end{align*} \] Now, apply the initial condition \( s(0) = 3 \): \[ s(0) = 3 \cos(0) + C = 3(1) + C = 3 + C = 3 \implies C = 0 \] **Final Position Function:** \[ s(t) = 3 \cos(\pi t) \] ### b. **Graph the Position Function on the Interval \( [0, 3] \).** The position function is: \[ s(t) = 3 \cos(\pi t) \] **Characteristics of the Graph:** - **Amplitude:** 3 meters - **Angular Frequency:** \( \pi \) radians/second - **Period:** \( \frac{2\pi}{\pi} = 2 \) seconds - **Phase Shift:** 0 - **Vertical Shift:** 0 **Key Points:** 1. **At \( t = 0 \):** \( s(0) = 3 \cos(0) = 3 \) meters (maximum point) 2. **At \( t = 1 \):** \( s(1) = 3 \cos(\pi \times 1) = -3 \) meters (minimum point) 3. **At \( t = 2 \):** \( s(2) = 3 \cos(2\pi) = 3 \) meters (maximum point) 4. **At \( t = 3 \):** \( s(3) = 3 \cos(3\pi) = -3 \) meters (minimum point) **Graph Sketch:** ``` s(t) | 3 +-----+-----+-----+-----+-----+-----+-----+ | * * * * | / \ / \ / \ / \ 0 +---+---+-----+---+-----+---+---+-----+---+-- t | / \ / \ / \ / \ / -3 +-----+-----+-----+-----+-----+-----+-----+ ``` - Peaks at \( t = 0, 2 \) - Troughs at \( t = 1, 3 \) ### c. **At What Times Does the Mass Reach Its Lowest Point the First Three Times?** The mass reaches its **lowest point** when \( s(t) \) is minimized. Given \( s(t) = 3 \cos(\pi t) \), the minimum value occurs when \( \cos(\pi t) = -1 \). \[ \cos(\pi t) = -1 \implies \pi t = \pi + 2\pi k \quad \text{for integer } k \geq 0 \] \[ t = 1 + 2k \quad \text{for integer } k \geq 0 \] **First Three Times:** 1. **First Minimum:** \( t = 1 + 2(0) = 1 \) second 2. **Second Minimum:** \( t = 1 + 2(1) = 3 \) seconds 3. **Third Minimum:** \( t = 1 + 2(2) = 5 \) seconds **So, the mass reaches its lowest point at \( t = 1 \) s, \( t = 3 \) s, and \( t = 5 \) s.** ### d. **At What Times Does the Mass Reach Its Highest Point the First Three Times?** The mass reaches its **highest point** when \( s(t) \) is maximized. Given \( s(t) = 3 \cos(\pi t) \), the maximum value occurs when \( \cos(\pi t) = 1 \). \[ \cos(\pi t) = 1 \implies \pi t = 0 + 2\pi k \quad \text{for integer } k \geq 0 \] \[ t = 2k \quad \text{for integer } k \geq 0 \] **First Three Times:** 1. **First Maximum:** \( t = 2(0) = 0 \) seconds 2. **Second Maximum:** \( t = 2(1) = 2 \) seconds 3. **Third Maximum:** \( t = 2(2) = 4 \) seconds **So, the mass reaches its highest point at \( t = 0 \) s, \( t = 2 \) s, and \( t = 4 \) s.**