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, we need to find the value of \( \frac{u^2}{Sg} \), where \( S \) is the displacement of the particle between \( t = \frac{7t_0}{8} \) and \( t = t_0 \).

### Step 1: Understand the motion of the particle
The particle is thrown vertically upward, and it reaches its maximum height at time \( t = t_0 \). At this point, its velocity is \( 0 \).

### Step 2: Calculate the maximum height \( H_{max} \)
Using the equation of motion:
\[ v = u - gt \]
At the maximum height, \( v = 0 \) and \( t = t_0 \):
\[ 0 = u - gt_0 \implies t_0 = \frac{u}{g} \]

Now, substituting \( t_0 \) into the equation for maximum height:
\[ H_{max} = ut_0 - \frac{1}{2}g(t_0^2) \]
Substituting \( t_0 = \frac{u}{g} \):
\[ H_{max} = u\left(\frac{u}{g}\right) - \frac{1}{2}g\left(\frac{u}{g}\right)^2 \]
\[ H_{max} = \frac{u^2}{g} - \frac{1}{2}g\left(\frac{u^2}{g^2}\right) = \frac{u^2}{g} - \frac{u^2}{2g} = \frac{u^2}{2g} \]

### Step 3: Calculate the height \( h \) at \( t = \frac{7t_0}{8} \)
Using the same equation of motion:
\[ h = u\left(\frac{7t_0}{8}\right) - \frac{1}{2}g\left(\frac{7t_0}{8}\right)^2 \]
Substituting \( t_0 = \frac{u}{g} \):
\[ h = u\left(\frac{7u}{8g}\right) - \frac{1}{2}g\left(\frac{7u}{8g}\right)^2 \]
\[ h = \frac{7u^2}{8g} - \frac{1}{2}g\left(\frac{49u^2}{64g^2}\right) = \frac{7u^2}{8g} - \frac{49u^2}{128g} \]
To combine these fractions, we convert \( \frac{7u^2}{8g} \) to a common denominator:
\[ \frac{7u^2}{8g} = \frac{112u^2}{128g} \]
Thus,
\[ h = \frac{112u^2}{128g} - \frac{49u^2}{128g} = \frac{63u^2}{128g} \]

### Step 4: Calculate the displacement \( S \)
The displacement \( S \) between \( t = \frac{7t_0}{8} \) and \( t = t_0 \) is given by:
\[ S = H_{max} - h = \frac{u^2}{2g} - \frac{63u^2}{128g} \]
To combine these, we convert \( \frac{u^2}{2g} \) to a common denominator:
\[ \frac{u^2}{2g} = \frac{64u^2}{128g} \]
Thus,
\[ S = \frac{64u^2}{128g} - \frac{63u^2}{128g} = \frac{u^2}{128g} \]

### Step 5: Calculate \( \frac{u^2}{Sg} \)
Now we can find \( \frac{u^2}{Sg} \):
\[ S = \frac{u^2}{128g} \implies gS = g \cdot \frac{u^2}{128g} = \frac{u^2}{128} \]
Thus,
\[ \frac{u^2}{Sg} = \frac{u^2}{\frac{u^2}{128}} = 128 \]

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