A solid sphere falls with a terminal velocity v in air .If it is allowed to fall in vaccum,

Options

A. terminal velocity of sphere = v
B. terminal velocity of sphere < v
C. terminal velocity of sphere > v
D. sphere never attains terminal velocity

Explanation

To solve the problem of a solid sphere falling with a terminal velocity \( v \) in air and determining what happens when it falls in a vacuum, we can follow these steps:

### Step 1: Understand Terminal Velocity in Air
The terminal velocity \( v_t \) of a sphere falling through a fluid is given by the formula:
\[ v_t = \frac{2}{9} \cdot \frac{r^2 g (\rho - \sigma)}{\eta} \]
where:
- \( r \) is the radius of the sphere,
- \( g \) is the acceleration due to gravity,
- \( \rho \) is the density of the sphere,
- \( \sigma \) is the density of the fluid (air in this case),
- \( \eta \) is the coefficient of viscosity of the fluid.

### Step 2: Analyze the Situation in Air
In air, the sphere experiences a drag force due to the viscosity of the air. This drag force balances the gravitational force acting on the sphere when it reaches terminal velocity \( v \). Therefore, the sphere falls with a constant speed \( v \).

### Step 3: Consider the Situation in Vacuum
When the sphere is allowed to fall in a vacuum:
- There are no air particles to exert a drag force on the sphere.
- The coefficient of viscosity \( \eta \) approaches zero because there is no fluid medium.

### Step 4: Evaluate the Terminal Velocity in Vacuum
Substituting \( \eta = 0 \) into the terminal velocity formula:
\[ v_t = \frac{2}{9} \cdot \frac{r^2 g (\rho - \sigma)}{0} \]
This expression indicates that the terminal velocity \( v_t \) approaches infinity, which implies that the sphere will continue to accelerate indefinitely without reaching a terminal velocity.

### Step 5: Conclusion
Since the sphere does not experience any viscous drag in a vacuum, it never attains a terminal velocity. Therefore, the correct answer is that the sphere never attains terminal velocity.

### Final Answer:
The correct option is: **The sphere never attains terminal velocity.**

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