A ball of mass m is dropped from a cliff of height H. The ratio of its kinetic energy to the potential energy when it is fallen through a height 3/4 H is

Options

A. `3:4`
B. `4:3`
C. `1:3`
D. `3:1`

Explanation

To solve the problem of finding the ratio of kinetic energy to potential energy for a ball of mass \( m \) dropped from a height \( H \) after it has fallen through a height of \( \frac{3}{4}H \), we can follow these steps:

### Step 1: Define the heights
Let:
- The initial height of the ball, \( H \).
- The height after falling through \( \frac{3}{4}H \) is \( h = H - \frac{3}{4}H = \frac{1}{4}H \).

### Step 2: Calculate the potential energy at height \( \frac{1}{4}H \)
The potential energy (PE) at this height is given by:
\[ PE = mgh = mg\left(\frac{1}{4}H\right) = \frac{mgH}{4} \]

### Step 3: Calculate the kinetic energy when the ball has fallen through \( \frac{3}{4}H \)
Using the conservation of energy, the total mechanical energy at the top (initial potential energy) is equal to the total mechanical energy at the height \( \frac{1}{4}H \) (sum of potential energy and kinetic energy).

Initial potential energy at height \( H \):
\[ PE_{\text{initial}} = mgH \]

At height \( \frac{1}{4}H \):
\[ PE + KE = mgH \]
Substituting the potential energy we found:
\[ \frac{mgH}{4} + KE = mgH \]
Now, solving for kinetic energy (KE):
\[ KE = mgH - \frac{mgH}{4} = mgH\left(1 - \frac{1}{4}\right) = mgH\left(\frac{3}{4}\right) = \frac{3mgH}{4} \]

### Step 4: Calculate the ratio of kinetic energy to potential energy
Now we can find the ratio of kinetic energy (KE) to potential energy (PE):
\[ \text{Ratio} = \frac{KE}{PE} = \frac{\frac{3mgH}{4}}{\frac{mgH}{4}} \]

### Step 5: Simplify the ratio
The \( mgH \) and \( \frac{1}{4} \) cancel out:
\[ \text{Ratio} = \frac{3}{1} = 3 \]

### Conclusion
The ratio of kinetic energy to potential energy when the ball has fallen through a height of \( \frac{3}{4}H \) is \( 3:1 \).