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

Options

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

Explanation

To solve the problem, we need to find the ratio of potential energy (PE) to kinetic energy (KE) of a stone projected vertically upwards at a height of \( \frac{4}{5}h \).

### Step-by-Step Solution:

1. **Identify the Maximum Height**:
- The stone is projected to reach a maximum height \( h \).

2. **Determine Total Energy**:
- At the maximum height \( h \), the potential energy (PE) is maximum and kinetic energy (KE) is zero.
- Total energy \( E \) at maximum height \( h \) is given by:
\[ E = PE = mgh \]

3. **Calculate Potential Energy at Height \( \frac{4}{5}h \)**:
- The potential energy at height \( \frac{4}{5}h \) is:
\[ PE = mg \left(\frac{4}{5}h\right) = \frac{4}{5}mgh \]

4. **Calculate Kinetic Energy at Height \( \frac{4}{5}h \)**:
- The total energy is conserved, so at height \( \frac{4}{5}h \):
\[ E = PE + KE \]
- Substituting the values we have:
\[ mgh = \frac{4}{5}mgh + KE \]
- Rearranging gives:
\[ KE = mgh - \frac{4}{5}mgh = \frac{1}{5}mgh \]

5. **Find the Ratio of Potential Energy to Kinetic Energy**:
- Now, we can find the ratio of potential energy to kinetic energy:
\[ \text{Ratio} = \frac{PE}{KE} = \frac{\frac{4}{5}mgh}{\frac{1}{5}mgh} \]
- Simplifying this ratio:
\[ \text{Ratio} = \frac{4/5}{1/5} = 4 \]

6. **Final Result**:
- The ratio of potential energy to kinetic energy at height \( \frac{4}{5}h \) is:
\[ \text{Ratio} = 4:1 \]