Question

Which one of the following figures represents the qualitative variation of absolute deceleration \(\left|\dfrac{dV}{dt}\right|\) with altitude \(h\) (measured from mean sea level) for a space vehicle undergoing a ballistic entry into the Earth's atmosphere?

• A. A
• B. B
• C. C
• D. D

Hint:

In ballistic entry, deceleration scales like \(\rho V^2\). Density rises as you descend, but speed falls—so expect a peak ("max-\(q\)") rather than a monotonic trend.

Answer 1

The Correct Option is D

Step 1: Drag deceleration model.
For a ballistic entry (no lift guidance), the magnitude of drag deceleration is \[ \left|\frac{dV}{dt}\right| \;\approx\; \frac{D}{m} = \frac{1}{2}\,\frac{\rho V^2 C_D A}{m} = \frac{\rho V^2}{2\,\beta}, \] where \(\rho(h)\) is air density, \(V\) is speed, and \(\beta=\dfrac{m}{C_D A}\) is the ballistic coefficient (constant for a given vehicle/attitude).

Step 2: Competing trends during entry.
- \(\rho(h)\): increases rapidly as altitude decreases (roughly \(\rho \propto e^{-h/H}\) with scale height \(H\)).
- \(V\): starts very large at high \(h\) and decreases due to drag as the vehicle descends. Thus \(|dV/dt| \propto \rho V^2\) is the product of an increasing factor \(\rho(h)\) and a decreasing factor \(V^2(h)\).

Step 3: Location of the maximum.
At very high \(h\): \(\rho\) is tiny \(\Rightarrow |dV/dt|\) small despite large \(V\).
As \(h\) decreases: \(\rho\) grows quickly while \(V\) is still high \(\Rightarrow |dV/dt|\) increases.
Deeper down: \(V\) has dropped substantially (large energy already dissipated) even though \(\rho\) is higher \(\Rightarrow\) the product \(\rho V^2\) reaches a maximum (the "max-\(q\)"/max-deceleration region) and then decreases. Therefore the qualitative curve must be bell-shaped: small at high altitude, increasing to a peak at mid–altitudes, and falling again toward lower altitudes.

Step 4: Eliminate wrong plots.
(A) and (C) show large deceleration at high altitude—contradicts tiny \(\rho\).
(B) shows deceleration growing toward the ground monotonically—ignores the strong reduction in \(V\).
(D) matches the expected peak at an intermediate \(h\).

Final Answer:
\[ \boxed{\text{(D) Bell-shaped }|dV/dt|\text{ vs. }h\text{ with a peak at mid-altitude}} \]