To determine the correct equation for gravitational potential energy (GPE), we first recall the formula for gravitational potential energy in classical mechanics. The gravitational potential energy (GPE) is given by:
[tex]\[ \text{GPE} = mgh \][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the object,
- [tex]\( g \)[/tex] is the acceleration due to gravity,
- [tex]\( h \)[/tex] is the height above the reference point.
Let's analyze the given options in the problem and compare each one to the correct formula of GPE:
A. [tex]\( \text{GPE} = ME + \frac{1}{2} mv^2 \)[/tex]
In this expression, ME typically stands for mechanical energy, and [tex]\(\frac{1}{2} mv^2\)[/tex] represents the kinetic energy. The formula incorrectly combines mechanical energy and kinetic energy, neither of which define the gravitational potential energy on its own. This option does not match the correct equation for GPE.
B. [tex]\( \text{GPE} = \frac{ME}{\frac{1}{2} mv^2} \)[/tex]
This expression represents the ratio of mechanical energy to kinetic energy. Gravitational potential energy is not a ratio involving mechanical and kinetic energies. Therefore, this option is also incorrect.
C. [tex]\( \text{GPE} = ME - \frac{1}{2} mv^2 \)[/tex]
This expression subtracts kinetic energy from mechanical energy. However, this does not correctly represent gravitational potential energy, which depends on mass, gravitational acceleration, and height. Thus, this option is incorrect as well.
D. [tex]\( \text{GPE} = \frac{1}{2} mv^2 - ME \)[/tex]
This expression subtracts mechanical energy from kinetic energy, which is not the proper formula for gravitational potential energy. Thus, this option does not correctly define GPE and is incorrect.
Since none of these options correctly represent the equation for gravitational potential energy given by [tex]\( \text{GPE} = mgh \)[/tex], the correct conclusion is that none of the provided options correctly define gravitational potential energy.
Therefore, the answer to the given question is:
[tex]\[
\boxed{-1}
\][/tex]
This indicates that none of the given options is correct for representing the gravitational potential energy of a system.
