A small drop of water falls from rest through a large height h in air, the final velocity is

Options

A. `prop sqrt( h ) `
B. ` prop h `
C. ` prop ( ( 1)/( h ))`
D. almost independent of h

Answer

To solve the problem of determining the final velocity of a small drop of water falling from rest through a large height \( h \) in air, we need to analyze the forces acting on the drop and the concept of terminal velocity.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The drop of water falls under the influence of gravity. Initially, it starts from rest, meaning its initial velocity \( u = 0 \).

2. **Forces Acting on the Drop**:
- As the drop falls, it experiences two main forces:
- The gravitational force \( F_g = mg \) acting downward, where \( m \) is the mass of the drop and \( g \) is the acceleration due to gravity.
- The drag force \( F_d \) due to air resistance acting upward. This force increases with the velocity of the drop.

3. **Terminal Velocity**:
- The drop will eventually reach a constant velocity known as terminal velocity \( v_t \). At this point, the drag force equals the gravitational force:
\[ mg = F_d \]
- The formula for terminal velocity \( v_t \) in a fluid is given by:
\[ v_t = \frac{2}{9} \cdot \frac{r^2 (\rho - \sigma) g}{\eta} \]
where:
- \( r \) is the radius of the drop,
- \( \rho \) is the density of the fluid (air),
- \( \sigma \) is the density of the drop (water),
- \( \eta \) is the viscosity of the fluid.

4. **Independence from Height**:
- From the formula of terminal velocity, we can observe that it does not contain the height \( h \). This indicates that the terminal velocity is independent of the height from which the drop falls.
- Therefore, as the drop falls from a large height, it will eventually reach this terminal velocity, and its final velocity will be equal to \( v_t \).

5. **Conclusion**:
- Since the terminal velocity is independent of height, we conclude that the final velocity of the drop when it reaches terminal velocity is almost independent of the height \( h \) from which it falls.

### Final Answer:
The final velocity of the drop is **almost independent of height \( h \)**.

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