Question

Terminal velocity of drop of radius 1 cm is 10 cm/sec. 64 such balls are combined to make a large drop. Find terminal velocity of this larger drop. :

• A. 160 cm/sec
• B. 140 cm/sec
• C. 180 cm/sec
• D. 150 cm/sec

Hint:

For problems involving coalescing drops (rain, mercury, etc.), always start with the conservation of volume to find the relationship between the radii of the small and large drops. Then, use the proportionality of the quantity in question (like terminal velocity, capacitance, potential) with the radius.

Answer 1

The Correct Option is A

Step 1: Understanding the physical idea
When a small spherical drop falls through a viscous medium, it finally attains terminal velocity when the downward effective weight is balanced by the viscous drag force. For identical material and same surrounding fluid, the terminal speed depends only on the size of the drop.

Step 2: Relation between terminal velocity and size
The viscous drag force on a sphere is proportional to its radius and velocity, while the effective weight is proportional to the volume of the sphere. At terminal velocity:
Effective weight ∝ viscous drag
r³ ∝ r × v
This gives:
v ∝ r²

Step 3: Finding the radius of the combined drop
Let the radius of each small drop be r. When 64 identical drops combine, volume is conserved.

Volume of one drop ∝ r³
Volume of large drop ∝ R³
So:
R³ = 64 r³
R = 4r

Step 4: Comparing terminal velocities
Since terminal velocity is proportional to the square of radius:
v′ / v = (R / r)²
v′ / v = (4r / r)² = 16

Given terminal velocity of one small drop:
v = 10 cm/s
v′ = 16 × 10 = 160 cm/s

Step 5: Final Answer
The terminal velocity of the larger drop is 160 cm/s.