A small iron ball falls from rest through a large height h into a water column, the final velocity

Options

A. Does not depend on h
B. Is proportional to `sqrt(h)`
C. Is uniform
D. All these statements are not correct

Answer

To solve the problem of a small iron ball falling from rest through a large height \( h \) into a water column and determining its final velocity, we can follow these steps:

### Step 1: Understand the Initial Conditions
The iron ball is released from rest at a height \( h \). Therefore, the initial velocity \( u = 0 \).

### Step 2: Analyze the Motion in Air
As the ball falls through the air, it accelerates due to gravity. The acceleration \( a \) is equal to \( g \) (acceleration due to gravity). The distance fallen in the air can be calculated using the equations of motion, but since we are interested in the final velocity when it enters the water, we can focus on the transition to the water.

### Step 3: Transition to Water
Once the ball enters the water, it experiences a buoyant force that opposes its weight. The forces acting on the ball in water are:
- Weight of the ball, \( W = mg \)
- Buoyant force, \( B = \rho_{liquid} V g \) (where \( V \) is the volume of the ball)

### Step 4: Determine Terminal Velocity
As the ball continues to fall through the water, it will eventually reach a point where the buoyant force equals the weight of the ball. At this point, the ball will stop accelerating and will fall with a constant velocity known as terminal velocity (\( V_t \)).

The formula for terminal velocity is given by:
\[ V_t = \frac{2}{9} \frac{r^2 g}{\eta} (\rho_{body} - \rho_{liquid}) \]
where:
- \( r \) is the radius of the ball
- \( g \) is the acceleration due to gravity
- \( \eta \) is the viscosity of the liquid
- \( \rho_{body} \) is the density of the ball
- \( \rho_{liquid} \) is the density of the liquid

### Step 5: Conclusion on Final Velocity
The final velocity of the ball when it reaches the terminal velocity does not depend on the height \( h \) from which it fell. Instead, it depends on the properties of the ball and the liquid. Thus, the final velocity is constant once terminal velocity is reached.

### Final Answer
The final velocity of the iron ball as it falls through the water is the terminal velocity \( V_t \), which is given by the formula above.