Certainly! Let’s carefully analyze and solve the problem step-by-step. --- # Problem Breakdown: - **Flow velocity, \( V 10 \, \text{/s} \)** (constant) - **Circulation, \( \Gamma = \, \text{}^2text{s} \)** - **Sink strength, \( m = -500 \, \text{m}2/\text{s} \)** (negative because it's a sink) - Both are located at the origin. - **Point \( P \) at \((x, y) = (3, 2) \)** - **Find the velocity magnitude at point \( P \).** --- # Step 1: Understanding the flow components The total flow at point \( P \) is composed of three parts: 1. **Uniform flow** \( \vec{V}_\infty \) in the x-direction with magnitude \( V = 10\, \text{m/s} \). 2. **Vortex flow** due to circulation \( \Gamma \). 3. **Sink flow** due to sink strength \( m \). --- # Step 2: Velocity due to uniform flow The uniform flow velocity vector: \[ \vec{V}_\infty = V \hat{i} = 10 \hat{i} \, \text{m/s} \] This contributes a constant velocity component in the x-direction. --- # Step 3: Velocity due to vortex (circulation \( \Gamma \)) For a vortex at the origin, the velocity at point \( (x, y) \) is: \[ \vec{V}_\text{vortex} = \frac{\Gamma}{2\pi r^2} \begin{bmatrix} -y \\ x \end{bmatrix} \] Where: - \( r = \sqrt{x^2 + y^2} \) Plugging in \( (x, y) = (3, 2) \): \[ r = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.606 \] Now, the vortex velocity components: \[ V_{x,\text{vortex}} = - \frac{\Gamma}{2\pi r^2} y \] \[ V_{y,\text{vortex}} = \frac{\Gamma}{2\pi r^2} x \] Calculate denominator: \[ 2\pi r^2 = 2 \times 3.1416 \times (13) \approx 81.68 \] Calculate numerator: \[ \frac{\Gamma}{2\pi r^2} \approx \frac{200}{81.68} \approx 2.447\, \text{m/s} \] Now, the components: \[ V_{x,\text{vortex}} = - 2.447 \times 2 = -4.894\, \text{m/s} \] \[ V_{y,\text{vortex}} = 2.447 \times 3 = 7.341\, \text{m/s} \] --- # Step 4: Velocity due to sink \( m \) For a sink at the origin, the velocity at \( (x, y) \) is: \[ \vec{V}_\text{sink} = - \frac{m}{2\pi r} \hat{r} \] The radial unit vector: \[ \hat{r} = \frac{1}{r} (x, y) \] Since \( m = -500 \): \[ V_{x,\text{sink}} = - \frac{-500}{2\pi r} \times \frac{x}{r} = \frac{500}{2\pi r^2} x \] \[ V_{y,\text{sink}} = \frac{500}{2\pi r^2} y \] Calculate: \[ 2\pi r^2 = 81.68 \text{ (from previous step)} \] \[ V_{x,\text{sink}} = \frac{500}{81.68} \times 3 \approx 6.11 \times 3 = 18.33\, \text{m/s} \] \[ V_{y,\text{sink}} = 6.11 \times 2 = 12.22\, \text{m/s} \] --- # Step 5: Sum all velocity components Total velocities: \[ V_{x,\text{total}} = V + V_{x,\text{vortex}} + V_{x,\text{sink}} = 10 - 4.894 + 18.33 \approx 23.44\, \text{m/s} \] \[ V_{y,\text{total}} = + 7.341 + 12.22 \approx 19.56\, \text{m/s} \] --- # Step 6: Calculate the magnitude of the velocity at \( P \) \[ V_{total} = \sqrt{V_{x,\text{total}}^2 + V_{y,\text{total}}^2} \] \[ V_{total} = \sqrt{(23.44)^2 + (19.56)^2} \approx \sqrt{549.8 + 382.4} = \sqrt{932.2} \approx 30.54\, \text{m/s} \] --- # **Final Answer:** \[ \boxed{ \text{Velocity magnitude at point } (3, 2) \text{ is approximately } \mathbf{30.54\, \text{m/s}} } \]