A stone is projected vertically upward to reach maximum height h. The ratio of its kinetic energy to its potential energy at a height `(4)/(5)`h, will be

Options

A. `5:4`
B. `4:5`
C. `1:4`
D. `4:1`

Answer

To solve the problem, we need to find the ratio of kinetic energy (K) to potential energy (U) of a stone projected vertically upward at a height of \( \frac{4}{5}h \), where \( h \) is the maximum height reached by the stone.

### Step-by-step Solution:

1. **Understanding the Energy Conservation**:
- When the stone is projected upwards, its total mechanical energy is conserved. The total mechanical energy (E) at the maximum height \( h \) is given by:
\[ E = K + U \]
- At the maximum height \( h \), the kinetic energy \( K \) is zero and the potential energy \( U \) is maximum:
\[ U = mgh \]
- Therefore, the total energy \( E \) is:
\[ E = mgh \]

2. **Calculating Potential Energy at Height \( \frac{4}{5}h \)**:
- At a height \( \frac{4}{5}h \), the potential energy \( U_P \) can be calculated as:
\[ U_P = mgh' = mg\left(\frac{4}{5}h\right) = \frac{4}{5}mgh \]

3. **Using Conservation of Energy**:
- According to the conservation of energy, the total energy at height \( \frac{4}{5}h \) is still \( mgh \):
\[ E = K_P + U_P \]
- Substituting the expression for potential energy:
\[ mgh = K_P + \frac{4}{5}mgh \]

4. **Solving for Kinetic Energy**:
- Rearranging the equation to find the kinetic energy \( K_P \):
\[ K_P = mgh - \frac{4}{5}mgh = \frac{1}{5}mgh \]

5. **Finding the Ratio of Kinetic Energy to Potential Energy**:
- Now, we have:
- Kinetic Energy \( K_P = \frac{1}{5}mgh \)
- Potential Energy \( U_P = \frac{4}{5}mgh \)
- The ratio of kinetic energy to potential energy is:
\[ \frac{K_P}{U_P} = \frac{\frac{1}{5}mgh}{\frac{4}{5}mgh} \]
- Simplifying this gives:
\[ \frac{K_P}{U_P} = \frac{1}{4} \]

### Final Answer:
The ratio of kinetic energy to potential energy at a height of \( \frac{4}{5}h \) is \( \frac{1}{4} \).

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