To determine the correct equation that represents how the total mechanical energy (ME) relates to its kinetic energy (KE) and gravitational potential energy (GPE), let us consider the principles of mechanical energy.
Mechanical energy in a system is the sum of its kinetic energy and its potential energy. Here, we specifically focus on gravitational potential energy. The kinetic energy (KE) of a body is given by the formula:
[tex]\[ KE = \frac{1}{2}mv^2 \][/tex]
where [tex]\( m \)[/tex] is the mass of the body and [tex]\( v \)[/tex] is its velocity.
On the other hand, gravitational potential energy (GPE) is given by:
[tex]\[ GPE = mgh \][/tex]
where [tex]\( m \)[/tex] is the mass of the body, [tex]\( g \)[/tex] is the acceleration due to gravity, and [tex]\( h \)[/tex] is the height of the body above the reference level.
The law of conservation of mechanical energy states that the total mechanical energy (ME) in a closed system remains constant if only conservative forces are acting. This implies that the total mechanical energy (ME) is the sum of the kinetic energy (KE) and the gravitational potential energy (GPE):
[tex]\[ ME = KE + GPE \][/tex]
So, let us analyze the given options:
A. [tex]\( ME = GPE - KE \)[/tex]: This option suggests subtracting kinetic energy from gravitational potential energy, which does not align with the conservation principle.
B. [tex]\( ME = KE - GPE \)[/tex]: This option suggests subtracting gravitational potential energy from kinetic energy, which is incorrect by the conservation principle.
c. [tex]\( ME = KE + GPE \)[/tex]: This option correctly expresses that the total mechanical energy is the sum of kinetic energy and gravitational potential energy.
D. [tex]\( ME = GPE \times KE \)[/tex]: This option suggests multiplying gravitational potential energy by kinetic energy, which is not a correct operation based on the conservation principle.
Therefore, the correct equation representing how the total mechanical energy (ME) of a system relates to its kinetic energy (KE) and gravitational potential energy (GPE) is:
[tex]\[ \boxed{ME = KE + GPE} \][/tex]
Hence, the correct answer is:
c. [tex]\( ME = KE + GPE \)[/tex].
