Question

Two rain drops of same radius '\(r\)' falling with same terminal velocity '\(V\)' merge and form bigger drop of radius '\(R\)'. The terminal velocity of big drop is

• A. \(\frac{VR^2}{r^2}\)
• B. \(\frac{2VR}{r}\)
• C. \(\frac{VR}{r}\)
• D. \(Vr^2/R^2\)

Hint:

Terminal velocity varies as the square of the radius, and volume is conserved on merging.

Answer 1

The Correct Option is A

Step 1: Terminal velocity law
\(v_T=\frac{2r^2(\rho-\sigma)g}{9\eta}\), so \(v_T\propto r^2\).

Step 2: Volume conservation
Two drops merge: \(\frac43\pi R^3=2\cdot\frac43\pi r^3\), so \(R^3=2r^3\).

Step 3: Result
\(\frac{V^{\prime}}{V}=\frac{R^2}{r^2}\), so \(V^{\prime}=\frac{VR^2}{r^2}\). Option (A).

Final Answer:
The terminal velocity is \(\frac{VR^2}{r^2}\), option (A). \[ \boxed{\text{(A)}} \]