A particle thrown up vertically from the ground at reaches its highest point at ` t=t_0 ` =The displacement of the particle between ` t=( 7t_0)/(8) and t=t_0` is S. Then ` (u^(2))/(Sg) = `

Explanation

To solve the problem step by step, we will analyze the motion of a particle thrown vertically upward and derive the required expression.

### Step-by-Step Solution

1. **Understanding the Motion**:
- A particle is thrown vertically upward with an initial velocity \( u \).
- It reaches its highest point at time \( t = t_0 \), where the final velocity \( v = 0 \).

2. **Finding the Time to Reach Maximum Height**:
- At the highest point, the velocity is zero. Using the equation of motion:
\[ v = u - gt \]
Setting \( v = 0 \) at \( t = t_0 \):
\[ 0 = u - gt_0 \implies t_0 = \frac{u}{g} \]

3. **Displacement Between Specific Times**:
- We need to find the displacement \( S \) between \( t = \frac{7t_0}{8} \) and \( t = t_0 \).
- The displacement can be calculated using the formula:
\[ S = h_{\text{max}} - h \]
- Here, \( h_{\text{max}} \) is the maximum height reached, and \( h \) is the height at \( t = \frac{7t_0}{8} \).

4. **Calculating Maximum Height \( h_{\text{max}} \)**:
- Using the equation:
\[ h_{\text{max}} = \frac{u^2}{2g} \]

5. **Finding Height at \( t = \frac{7t_0}{8} \)**:
- Substitute \( t = \frac{7t_0}{8} \) into the displacement formula:
\[ h = ut - \frac{1}{2}gt^2 \]
- Substitute \( t_0 = \frac{u}{g} \):
\[ h = u \left(\frac{7u}{8g}\right) - \frac{1}{2}g \left(\frac{7u}{8g}\right)^2 \]
- Simplifying:
\[ h = \frac{7u^2}{8g} - \frac{1}{2}g \cdot \frac{49u^2}{64g^2} \]
\[ h = \frac{7u^2}{8g} - \frac{49u^2}{128g} \]
- Finding a common denominator:
\[ h = \frac{112u^2}{128g} - \frac{49u^2}{128g} = \frac{63u^2}{128g} \]

6. **Calculating Displacement \( S \)**:
- Now substitute \( h_{\text{max}} \) and \( h \) into the displacement formula:
\[ S = h_{\text{max}} - h = \frac{u^2}{2g} - \frac{63u^2}{128g} \]
- Finding a common denominator:
\[ S = \frac{64u^2}{128g} - \frac{63u^2}{128g} = \frac{u^2}{128g} \]

7. **Finding the Required Ratio**:
- We need to find \( \frac{u^2}{Sg} \):
\[ \frac{u^2}{Sg} = \frac{u^2}{\left(\frac{u^2}{128g}\right)g} = \frac{u^2 \cdot 128g}{u^2g} = 128 \]

### Final Answer:
\[ \frac{u^2}{Sg} = 128 \]