Question

'\(n\)' identical small drops of water are falling down vertically through a medium each with a terminal velocity '\(v\)'. If they combine to form a single drop, then its terminal velocity is \(xv\). The value of \(x\) is

• A. \(n^{1/3}\)
• B. \(n^{2/3}\)
• C. \(n^{2}\)
• D. \(n^{3}\)

Hint:

Terminal velocity varies as the square of the radius. Equal volume gives the radius of the large drop.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Terminal velocity of a drop is \(v\propto r^{2}\). When drops merge, the total volume stays the same.

Step 2: Radius of Big Drop
\(\dfrac{4}{3}\pi R^{3}=n\cdot\dfrac{4}{3}\pi r^{3}\)
So \(R=n^{1/3}r\).

Step 3: Calculation
\(\dfrac{v'}{v}=\dfrac{R^{2}}{r^{2}}=n^{2/3}\)
So \(x=n^{2/3}\).

Step 4: Compare the options
This is the second option.

Final Answer:
The terminal velocity becomes n to the power 2/3 times v. \[ \boxed{\text{(B) }n^{2/3}} \]