To determine which equation correctly represents the gravitational potential energy (GPE) of a system, let's review the concept of GPE and then analyze each of the given options.
Gravitational Potential Energy (GPE):
The formula for gravitational potential energy is:
[tex]\[ \text{GPE} = m \cdot g \cdot h \][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the object,
- [tex]\( g \)[/tex] is the acceleration due to gravity (approximately [tex]\( 9.8 \text{ m/s}^2 \)[/tex] on the surface of the Earth), and
- [tex]\( h \)[/tex] is the height of the object above the reference point.
This formula indicates that GPE is directly proportional to the mass, the height of the object, and the acceleration due to gravity.
Now let's examine each of the provided choices:
### Option A:
[tex]\[ \text{GPE} = ME + \frac{1}{2} mv^2 \][/tex]
This option combines mechanical energy (ME) with the kinetic energy term. This equation is incorrect because GPE does not involve kinetic energy ([tex]\(\frac{1}{2}mv^2\)[/tex]) or the entire mechanical energy (ME).
### Option B:
[tex]\[ \text{GPE} = \frac{ME}{\frac{1}{2} mv^2} \][/tex]
This option represents GPE as the division of mechanical energy by the kinetic energy term. This is also incorrect because GPE is not derived from a ratio involving mechanical energy and kinetic energy.
### Option C:
[tex]\[ \text{GPE} = ME - \frac{1}{2} mv^2 \][/tex]
This option tries to represent GPE as a difference between mechanical energy and the kinetic energy term. This is incorrect because GPE does not involve subtracting the kinetic energy from the mechanical energy.
### Option D:
[tex]\[ \text{GPE} = \frac{1}{2} mv^2 - ME \][/tex]
This option represents GPE as the difference between the kinetic energy term and the mechanical energy. This is also incorrect because it does not follow the proper relationship for gravitational potential energy.
Conclusion:
None of the provided options correctly represent the gravitational potential energy ([tex]\(\text{GPE} = mgh\)[/tex]) using the given formulas. After carefully analyzing each option:
- A incorrectly combines ME and kinetic energy,
- B incorrectly represents a ratio involving ME and kinetic energy,
- C incorrectly subtracts kinetic energy from ME,
- D incorrectly subtracts ME from kinetic energy.
Thus, none of these formulas correctly match the true equation for GPE.
### Final Answer:
None of the provided equations correctly represent gravitational potential energy.
