During upwards and downwards journeys under gravity, a body passes through a point at 2 s and 14 s. The total height from ground is

Options

A. 4g
B. 8g
C. 16g
D. 32g

Explanation

To solve the problem, we need to determine the total height from the ground that the body reaches during its upward and downward journey under gravity. We know that the body passes through a certain point at 2 seconds and 14 seconds.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The body is thrown upwards and then comes back down. It passes a certain point (let's call it point C) at 2 seconds during its upward journey and again at 14 seconds during its downward journey.

2. **Calculating the Time Interval**:
- The time taken from the point C (2 seconds) to the point C' (14 seconds) is:
\[ \text{Total time} = 14 \text{ s} - 2 \text{ s} = 12 \text{ s} \]
- This 12 seconds includes the time taken to reach the maximum height (B) and then come back down to point C'.

3. **Dividing the Time**:
- Since the motion is symmetrical, the time taken to go from C to B (upwards) is equal to the time taken to go from B' to C' (downwards). Therefore, the time taken to reach the maximum height (B) is:
\[ \text{Time to reach B} = \frac{12 \text{ s}}{2} = 6 \text{ s} \]

4. **Total Time of Flight**:
- The total time from the starting point A to the maximum height B and back down to the ground is:
\[ \text{Total time from A to B and B to ground} = 2 \text{ s} + 6 \text{ s} + 6 \text{ s} = 14 \text{ s} \]
- The total time from A to B and back to the ground is 8 seconds (6 seconds to reach B and 2 seconds to go from C' to the ground).

5. **Using the Equation of Motion**:
- We can use the equation of motion to find the height (S):
\[ S = ut + \frac{1}{2} a t^2 \]
- Here, \(u\) is the initial velocity, \(a\) is the acceleration due to gravity (which is \(-g\)), and \(t\) is the total time of 8 seconds.
- Since the body reaches the maximum height at point B, the final velocity at that point is 0. Therefore, we can consider the motion from the maximum height back to the ground.

6. **Calculating the Height**:
- The height can be calculated using the time taken to reach the maximum height (6 seconds):
\[ S = 0 \cdot 6 + \frac{1}{2} g (6^2) = \frac{1}{2} g \cdot 36 = 18g \]
- However, we need to account for the entire journey (up and down). The total height from A to B is:
\[ S = \frac{1}{2} g (8^2) = \frac{1}{2} g \cdot 64 = 32g \]

7. **Final Result**:
- The total height from the ground is:
\[ \text{Total height} = 32g \]