In the above problem if body is thrown down with velocity 'u' equation for the descent time is

Options

A. `h = (1)/(2) gt^(2)`
B. `h =ut + (1)/(2) gt^(2)`
C. `-h = -ut + (1)/(2) gt^(2)`
D. `-h = ut + (1)/(2)gt^(2)`

Answer

To solve the problem of finding the descent time for a body thrown downward with an initial velocity \( u \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A body is thrown downward from a height \( h \) with an initial velocity \( u \).
- We need to find the time \( t \) it takes for the body to reach the ground.

2. **Setting Up the Equation of Motion**:
- We will use the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
- Here, \( s \) is the displacement, \( u \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time.

3. **Choosing the Sign Convention**:
- We will take the downward direction as negative:
- Displacement \( s = -h \) (since it moves downward)
- Initial velocity \( u = -u \) (since it is thrown downward)
- Acceleration due to gravity \( a = -g \) (since it acts downward)

4. **Substituting Values into the Equation**:
- Plugging the values into the equation:
\[ -h = -u t - \frac{1}{2} g t^2 \]
- Rearranging gives:
\[ h = ut + \frac{1}{2} g t^2 \]

5. **Rearranging the Equation**:
- Rearranging the equation to form a standard quadratic equation:
\[ \frac{1}{2} g t^2 + ut - h = 0 \]

6. **Applying the Quadratic Formula**:
- This is a quadratic equation in the form \( at^2 + bt + c = 0 \), where:
- \( a = \frac{1}{2} g \)
- \( b = u \)
- \( c = -h \)
- Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ t = \frac{-u \pm \sqrt{u^2 - 4 \cdot \frac{1}{2} g \cdot (-h)}}{2 \cdot \frac{1}{2} g} \]
- Simplifying:
\[ t = \frac{-u \pm \sqrt{u^2 + 2gh}}{g} \]

7. **Choosing the Positive Root**:
- Since time cannot be negative, we take the positive root:
\[ t = \frac{-u + \sqrt{u^2 + 2gh}}{g} \]

### Final Answer:
The equation for the descent time \( t \) when the body is thrown downward with an initial velocity \( u \) is:
\[ t = \frac{-u + \sqrt{u^2 + 2gh}}{g} \]