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 of finding the ratio of kinetic energy (KE) to potential energy (PE) of a stone projected vertically upward at a height of \( \frac{4}{5}h \), we can follow these steps:

### Step 1: Understand the energies at maximum height
When the stone reaches its maximum height \( h \), all its initial kinetic energy has been converted into potential energy. At this point, the potential energy (PE) is given by:
\[ PE = mgh \]
where \( m \) is the mass of the stone, \( g \) is the acceleration due to gravity, and \( h \) is the maximum height.

### Step 2: Determine the potential energy at height \( \frac{4}{5}h \)
At a height of \( \frac{4}{5}h \), the potential energy (PE) is:
\[ PE = mg\left(\frac{4}{5}h\right) = \frac{4}{5}mgh \]

### Step 3: Use conservation of energy
The total mechanical energy at the initial point (ground level) is equal to the total mechanical energy at any point during the motion. At the height \( \frac{4}{5}h \), the total energy is still \( mgh \). Therefore, we can write:
\[ \text{Total Energy} = KE + PE \]
Substituting the values we have:
\[ mgh = KE + \frac{4}{5}mgh \]

### Step 4: Solve for kinetic energy (KE)
Rearranging the equation to find the kinetic energy (KE):
\[ KE = mgh - \frac{4}{5}mgh = mgh\left(1 - \frac{4}{5}\right) = mgh\left(\frac{1}{5}\right) \]

### Step 5: Calculate the ratio of KE to PE
Now, we have both kinetic energy and potential energy at height \( \frac{4}{5}h \):
- Kinetic Energy (KE) = \( \frac{1}{5}mgh \)
- Potential Energy (PE) = \( \frac{4}{5}mgh \)

The ratio of kinetic energy to potential energy is:
\[ \frac{KE}{PE} = \frac{\frac{1}{5}mgh}{\frac{4}{5}mgh} \]

### Step 6: Simplify the ratio
Cancelling \( mgh \) from the numerator and denominator, we get:
\[ \frac{KE}{PE} = \frac{1}{5} \div \frac{4}{5} = \frac{1}{4} \]

### Final Answer
Thus, the ratio of kinetic energy to potential energy at a height of \( \frac{4}{5}h \) is:
\[ \frac{KE}{PE} = \frac{1}{4} \]