Let's break down the problem **step-by-step**: --- ## **Step 1: Extract Given Values** - \( r = 17 \) ft (constant) - \( \theta = 0.25t \) rad - \( z = 1.8\sin(\theta) \) ft - \( t = 15 \) s --- ## **Step 2: Cylindrical Coordinates to Velocity and Acceleration** ### **General Formulas in Cylindrical Coordinates** For position: \( \vec{r} = r\hat{u}_r + z \hat{k} \) With \( r, \theta, z \) all functions of time: **Velocity:** \[ \vec{v} = \dot{r}\hat{u}_r + r\dot{\theta}\hat{u}_\theta + \dot{z}\hat{k} \] **Acceleration:** \[ \vec{a} = (\ddot{r} - r\dot{\theta}^2)\hat{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{u}_\theta + \ddot{z}\hat{k} \] But since \( r \) is constant: - \( \dot{r} = 0 \) - \( \ddot{r} = 0 \) So, \[ \vec{v} = r\dot{\theta}\hat{u}_\theta + \dot{z}\hat{k} \] \[ \vec{a} = (-r\dot{\theta}^2)\hat{u}_r + (r\ddot{\theta})\hat{u}_\theta + \ddot{z}\hat{k} \] --- ## **Step 3: Compute Needed Quantities at \( t = 15 \) s** ### **Calculate \( \theta \), \( \dot{\theta} \), \( \ddot{\theta} \):** \[ \theta = 0.25t \implies \dot{\theta} = 0.25, \quad \ddot{\theta} = 0 \] So: - \( \theta(15) = 0.25 \times 15 = 3.75 \) rad - \( \dot{\theta} = 0.25 \text{ rad/s} \) - \( \ddot{\theta} = 0 \) ### **Calculate \( z \), \( \dot{z} \), \( \ddot{z} \):** \[ z = 1.8\sin(\theta) \] \[ \dot{z} = 1.8\cos(\theta)\dot{\theta} \] \[ \ddot{z} = 1.8[-\sin(\theta)\dot{\theta}^2 + \cos(\theta)\ddot{\theta}] \] But \( \ddot{\theta}=0 \), so: \[ \ddot{z} = -1.8\sin(\theta)\dot{\theta}^2 \] Now substitute \( \theta = 3.75 \), \( \dot{\theta}=0.25 \): - \( \sin(3.75) \approx -0.5716 \) - \( \cos(3.75) \approx -0.8206 \) \[ z = 1.8 \times (-0.5716) = -1.0289 \text{ ft} \] \[ \dot{z} = 1.8 \times (-0.8206) \times 0.25 = -0.3693 \text{ ft/s} \] \[ \ddot{z} = -1.8 \times (-0.5716) \times (0.25)^2 = 1.8 \times 0.5716 \times 0.0625 = 0.0643 \text{ ft/s}^2 \] --- ## **Step 4: Plug Into Velocity and Acceleration Formulas** ### **Velocity Vector** \[ \vec{v} = r\dot{\theta} \hat{u}_\theta + \dot{z} \hat{k} \] \[ r\dot{\theta} = 17 \times 0.25 = 4.25 \text{ ft/s} \] So: \[ \vec{v}(15s) = 0\,\hat{u}_r + 4.25\,\hat{u}_\theta + (-0.37)\,\hat{k} \approx 0\,\hat{u}_r + 4.25\,\hat{u}_\theta - 0.37\,\hat{k} \] ### **Acceleration Vector** \[ \vec{a} = (-r\dot{\theta}^2)\hat{u}_r + (r\ddot{\theta})\hat{u}_\theta + \ddot{z}\hat{k} \] \[ r\dot{\theta}^2 = 17 \times (0.25)^2 = 17 \times 0.0625 = 1.0625 \] So: \[ \vec{a}(15s) = -1.06\,\hat{u}_r + 0\,\hat{u}_\theta + 0.064\,\hat{k} \] --- ## **Step 5: Magnitudes** ### **Velocity Magnitude** \[ |\vec{v}| = \sqrt{(4.25)^2 + (-0.37)^2} = \sqrt{18.0625 + 0.1369} = \sqrt{18.1994} \approx 4.27\, \text{ft/s} \] ### **Acceleration Magnitude** \[ |\vec{a}| = \sqrt{(-1.06)^2 + (0)^2 + (0.064)^2} = \sqrt{1.1236 + 0.0041} = \sqrt{1.1277} \approx 1.06\, \text{ft/s}^2 \] --- ## **Step 6: Final Answers (with units!)** ### **Vectors** \[ \boxed{ \vec{v}(15s) = 0\,\hat{u}_r + 4.25\,\hat{u}_\theta - 0.37\,\hat{k} \,\,\, \text{(ft/s)} } \] \[ \boxed{ \vec{a}(15s) = -1.06\,\hat{u}_r + 0\,\hat{u}_\theta + 0.064\,\hat{k} \,\,\, \text{(ft/s}^2\text{)} } \] ### **Magnitudes** \[ \boxed{ |\vec{v}(15s)| = 4.27\, \text{ft/s} } \] \[ \boxed{ |\vec{a}(15s)| = 1.06\, \text{ft/s}^2 } \] --- ### **Summary Table** | Quantity | Value | |-------------------------|------------------------------------| | \(\vec{v}(15s)\) | \(0\,\hat{u}_r + 4.25\,\hat{u}_\theta - 0.37\,\hat{k}\) ft/s | | \(\vec{a}(15s)\) | \(-1.06\,\hat{u}_r + 0\,\hat{u}_\theta + 0.064\,\hat{k}\) ft/s² | | \(|\vec{v}(15s)|\) | \(4.27\) ft/s | | \(|\vec{a}(15s)|\) | \(1.06\) ft/s² | Let me know if you need *any step* explained in more detail!