The acceleration of a sphere falling through a liquid is `(30 -3v) cm//s^(2)` where v is its speed in cm/s . The maximum possible velocity of the sphere and the time when it is achieved are

Options

A. 10 cm /s after 10 second
B. 10 cm/s instantly
C. 10 cm/s , will never be achieved
D. 30 cm/s , after 30 second

Answer

To solve the problem, we need to determine the maximum possible velocity of a sphere falling through a liquid, given its acceleration as \( a = 30 - 3v \) cm/s², where \( v \) is the velocity in cm/s. We also need to find the time when this maximum velocity is achieved.

### Step-by-step Solution:

1. **Understanding the Acceleration**:
The acceleration of the sphere is given by:
\[ a = 30 - 3v \]
We know that acceleration can also be expressed as the change in velocity with respect to time:
\[ a = \frac{dv}{dt} \]
Therefore, we can equate the two expressions:
\[ \frac{dv}{dt} = 30 - 3v \]

2. **Rearranging the Equation**:
We can rearrange the equation to isolate \( dv \) and \( dt \):
\[ \frac{dv}{30 - 3v} = dt \]

3. **Integrating Both Sides**:
We will integrate both sides. The left side requires a substitution. We can factor out the constant:
\[ \frac{1}{3} \int \frac{dv}{10 - v} = \int dt \]
Integrating gives:
\[ -\frac{1}{3} \ln |10 - v| = t + C \]
where \( C \) is the constant of integration.

4. **Solving for \( v \)**:
Rearranging the equation to solve for \( v \):
\[ \ln |10 - v| = -3t - 3C \]
Exponentiating both sides:
\[ |10 - v| = e^{-3t - 3C} \]
Let \( K = e^{-3C} \), then:
\[ 10 - v = \frac{K}{e^{3t}} \]
Thus:
\[ v = 10 - \frac{K}{e^{3t}} \]

5. **Finding Maximum Velocity**:
The maximum velocity occurs when the acceleration is zero:
\[ 30 - 3v = 0 \]
Solving for \( v \):
\[ 3v = 30 \implies v = 10 \text{ cm/s} \]

6. **Determining Time for Maximum Velocity**:
To find the time when this maximum velocity is achieved, we need to analyze the equation:
\[ 10 - v = \frac{K}{e^{3t}} \]
When \( v = 10 \):
\[ 10 - 10 = \frac{K}{e^{3t}} \implies 0 = \frac{K}{e^{3t}} \]
Since \( K \) is a constant, this implies that \( K \) must be zero for this equation to hold true, which means that the maximum velocity of 10 cm/s is never actually achieved in finite time.

### Conclusion:
- The maximum possible velocity of the sphere is \( 10 \) cm/s.
- This maximum velocity is never actually achieved in finite time.