A semi-infinite straight vortex filament of circulation $\Gamma = 50\text{ m}^2/\text{s}$ extends along the positive $x$-axis from $x = 0$ to $x = +\infty$. The induced velocity $w$ (in $\text{m/s}$) at a point located at $(0, 2\text{ m}, 0)$ in the plane perpendicular to the filament is closest to:

(A) 1.99 m/s

(B) 3.98 m/s

(C) 0.99 m/s

(D) 7.96 m/s

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles From the Biot-Savart law, the velocity induced by a straight vortex segment of circulation $\Gamma$ at a perpendicular distance $h$ is: $$w = \frac{\Gamma}{4\pi h} (\cos \theta_1 - \cos \theta_2)$$ For a semi-infinite straight vortex line extending from $x = 0$ to $x = +\infty$ with the point located directly opposite the origin: - $\theta_1 = 90^\circ \implies \cos \theta_1 = 0$ - $\theta_2 = 180^\circ \implies \cos \theta_2 = -1$ $$w = \frac{\Gamma}{4\pi h} (0 - (-1)) = \frac{\Gamma}{4\pi h}$$ Notice this is exactly half the velocity induced by an infinitely long straight vortex filament ($w_{\infty} = \frac{\Gamma}{2\pi h}$). ### 2. Step-by-Step Calculation Substitute known parameters: - $\Gamma = 50\text{ m}^2/\text{s}$ - $h = 2.0\text{ m}$ $$w = \frac{50}{4\pi \times 2.0} = \frac{50}{8\pi} = \frac{25}{4\pi} \approx \frac{25}{12.56637} \approx 1.9894\text{ m/s} \approx 1.99\text{ m/s}$$ ### 3. Distractor Forensics - **Option A (1.99 m/s):** Correct semi-infinite induced velocity $\frac{\Gamma}{4\pi h}$. - **Option B (3.98 m/s):** Evaluated as infinite vortex line $\frac{\Gamma}{2\pi h}$. - **Option C (0.99 m/s):** Extra factor of 2 in denominator. - **Option D (7.96 m/s):** Inverted factor or omitted $\pi$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Always verify whether the vortex is infinite ($\frac{\Gamma}{2\pi h}$) or semi-infinite ($\frac{\Gamma}{4\pi h}$). - **Standard Reference:** Anderson, J.D., *Fundamentals of Aerodynamics*, Chapter 5 (Incompressible Flow over Finite Wings).

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