Question

Two liquid drops of equal radii are falling through air with the terminal velocity \( v \). If these two drops coalesce to form a single drop, its terminal velocity will be

• A. \( \sqrt{2} \, v \)
• B. \( 2v \)
• C. \( \sqrt[3]{4} \, v \)
• D. \( \sqrt{2} \, v \)

Hint:

The terminal velocity of a liquid drop depends on the square of its radius. When drops coalesce, the radius increases by a factor of \( 2^{1/3} \), and so the terminal velocity increases by a factor of \( \sqrt[3]{4} \).

Answer 1

The Correct Option is C

The terminal velocity \( v_t \) of a liquid drop is proportional to the square of its radius \( r \), as given by: \[ v_t \propto r^2 \] When two drops of equal radii coalesce, their combined radius becomes \( r_{{new}} = 2^{1/3} r \). The new terminal velocity \( v_{{new}} \) will be proportional to the square of the new radius: \[ v_{{new}} \propto (2^{1/3} r)^2 = 2^{2/3} v \] Thus, the new terminal velocity will be \( \sqrt[3]{4} v \). Hence, the correct answer is (c).