During vertical motion under gravity a body passes through a point at 4s and 12 s in upwards and downwards journeys. The speed at this point is

Options

A. g
B. 2g
C. 3g
D. 4g

Explanation

To solve the problem of finding the speed of a body at a specific point during its vertical motion under gravity, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion**:
The body is projected upwards and then comes back down. It passes through a point (let's call it point C) at two different times: 4 seconds on the way up and 12 seconds on the way down.

2. **Time Intervals**:
The time taken to reach point C on the way up is 4 seconds. The time taken to reach point C on the way down is 12 seconds. The total time taken from the start to the point of maximum height (point B) and back down to point C is 12 seconds.

3. **Calculating the Time for Descent**:
The time taken to go from point C to point B (maximum height) and then back down to point C is:
\[ \text{Total time from C to B and B to C'} = 12 \text{ seconds} - 4 \text{ seconds} = 8 \text{ seconds} \]
This means the time taken to go from point C to point B (upwards) and back down to point C (downwards) is 8 seconds.

4. **Dividing the Time**:
Since the motion is symmetrical, the time taken to go from point C to point B (upwards) is equal to the time taken to go from point B back down to point C (downwards). Therefore, each segment (C to B and B to C) takes:
\[ \frac{8 \text{ seconds}}{2} = 4 \text{ seconds} \]

5. **Using the Equation of Motion**:
We can use the equation of motion to find the speed at point C when the body is descending:
\[ V = U + at \]
Here, \(U\) (initial velocity at point B) is 0 (since it is at maximum height), \(a\) is the acceleration due to gravity \(g\), and \(t\) is the time taken to fall from point B to point C, which we found to be 4 seconds.

6. **Substituting Values**:
Substituting the values into the equation:
\[ V = 0 + g \cdot 4 = 4g \]

7. **Final Answer**:
Therefore, the speed of the body at point C is:
\[ V = 4g \]

### Conclusion:
The speed at point C, where the body passes at 4 seconds on the way up and 12 seconds on the way down, is \(4g\).