Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.48)**
**1. Intuitive Physical Setup & Given Parameters**:
The question asks about the capabilities and limitations of the **single horseshoe vortex model** (bound vortex of span $b$ with two trailing semi-infinite vortices) in finite wing aerodynamic theory (MSQ).
**2. Step-by-Step Aerodynamic Derivation**:
1. **Single Horseshoe Vortex Limitations**:
- A single horseshoe vortex assumes uniform circulation $\Gamma$ across the span, which gives infinite downwash at the wing tips and fails to capture the realistic spanwise circulation distribution $\Gamma(y)$.
- Because induced drag is fundamentally related to the integral of downwash across the wing ($C_{Di} = \frac{1}{V_\infty S} \int \Gamma w dy$), a single horseshoe vortex **cannot** accurately calculate the wing's induced drag coefficient (it predicts infinity or requires ad-hoc cutoffs). Statement B is **FALSE**.
- Similarly, it models the chord as a single bound line, so it cannot capture chordwise pressure distribution or wing pitching moment $C_m$. Statement A is **FALSE**.
2. **Valid Applications of Single Horseshoe Vortex**:
- In the **far field** (at distances much larger than the chord, such as at the horizontal tail or behind the aircraft):
- The downwash induced by the main wing at the horizontal tail location can be very well approximated by treating the wing as a single horseshoe vortex placed at the aerodynamic center. Statement C is **TRUE**.
- In **formation flight**, the upwash field experienced by a trailing wing flying outboard and behind the lead aircraft's wingtip is dominated by the trailing vortex pair, which is accurately captured by the horseshoe model. Statement D is **TRUE**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) It can approximate the wing pitching moment coefficient**: ❌ **INCORRECT**. Requires chordwise vortex distribution, not a single spanwise line.
- **(B) It can approximate the wing induced drag coefficient**: ❌ **INCORRECT**. Produces singular/inaccurate self-induced downwash.
- **(C) It can approximate the effect of the wing on the induced drag coefficient of a typical horizontal tail**: ✅ **CORRECT**. Far-field downwash at the tailplane is well-predicted.
- **(D) It can approximate the aerodynamic benefit/penalty of formation flight compared to isolated flight**: ✅ **CORRECT**. Outboard upwash fields from trailing vortices are accurately captured.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C, D**
- **Exam Takeaway**: A single horseshoe vortex is invalid for self-induced near-field properties (induced drag, pitch moment) but highly effective for far-field aerodynamic interactions (tail downwash, formation flying upwash).