During free fall the total energy at 3/4th the height is .....

Options

A. constant
B. zero
C. gravitational potential energy at the top
D. 3/4th the initial potential energy.

Explanation

To solve the question regarding the total energy during free fall at three-fourths the height, we need to understand the concepts of potential energy and kinetic energy.

### Step-by-Step Solution:

1. **Understanding Total Energy**:
- The total mechanical energy (E) of an object in free fall is the sum of its gravitational potential energy (PE) and kinetic energy (KE).
- Mathematically, this can be expressed as:
\[ E = PE + KE \]

2. **Potential Energy at Height**:
- The potential energy (PE) of an object at a height \( h \) is given by the formula:
\[ PE = mgh \]
- Here, \( m \) is the mass of the object, \( g \) is the acceleration due to gravity, and \( h \) is the height from the ground.

3. **Total Energy at the Top**:
- At the top of the fall (height \( h \)), the object has maximum potential energy and zero kinetic energy (since it is at rest).
- Therefore, the total energy at the top is:
\[ E = mgh + 0 = mgh \]

4. **Free Fall to Three-Fourths Height**:
- When the object falls to three-fourths of the height, the height \( h' \) is:
\[ h' = \frac{3}{4}h \]
- The potential energy at this height is:
\[ PE' = mg\left(\frac{3}{4}h\right) = \frac{3}{4}mgh \]

5. **Kinetic Energy at Three-Fourths Height**:
- As the object falls, its potential energy is converted into kinetic energy. At three-fourths the height, the total energy remains constant (as established earlier).
- Thus, the kinetic energy (KE) at this height can be calculated as:
\[ KE' = E - PE' = mgh - \frac{3}{4}mgh = \frac{1}{4}mgh \]

6. **Total Energy at Three-Fourths Height**:
- The total energy at three-fourths of the height is still equal to the total energy at the top:
\[ E = mgh \]

### Conclusion:
The total energy during free fall at three-fourths the height is constant and equal to \( mgh \).

### Answer:
The correct option is that the total energy at three-fourths the height is **constant**.

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