To determine which equation correctly represents the relationship between the total mechanical energy (ME), kinetic energy (KE), and gravitational potential energy (GPE) of a system, let's first define each term:
1. Total Mechanical Energy (ME): The sum of the kinetic and potential energies in a system.
2. Kinetic Energy (KE): The energy that an object possesses due to its motion. Mathematically, [tex]\( KE = \frac{1}{2}mv^2 \)[/tex], where [tex]\(m\)[/tex] is mass and [tex]\(v\)[/tex] is velocity.
3. Gravitational Potential Energy (GPE): The energy an object possesses due to its position in a gravitational field. It is given by [tex]\( GPE = mgh \)[/tex], where [tex]\(m\)[/tex] is mass, [tex]\(g\)[/tex] is the acceleration due to gravity, and [tex]\(h\)[/tex] is height.
The principle of conservation of mechanical energy states that in the absence of non-conservative forces (like friction and air resistance), the total mechanical energy of a system remains constant. It is the sum of kinetic energy and potential energy.
In mathematical terms, the total mechanical energy (ME) is given by:
[tex]\[ ME = KE + GPE \][/tex]
Now, let's evaluate each of the given equations:
A. [tex]\( ME = KE - GPE \)[/tex]
- This equation implies that mechanical energy is the difference between kinetic and potential energy, which is incorrect based on the principle of conservation of mechanical energy.
B. [tex]\( ME = GPE - KE \)[/tex]
- This equation also implies a difference, which is not correct in this context.
C. [tex]\( ME = GPE \times KE \)[/tex]
- This equation suggests a product of kinetic and potential energy, which does not align with the correct representation of mechanical energy.
D. [tex]\( ME = KE + GPE \)[/tex]
- This equation states that the total mechanical energy is the sum of kinetic and potential energy, which is consistent with the principle of conservation of mechanical energy.
Therefore, the correct equation is:
[tex]\[ \boxed{ME = KE + GPE} \][/tex]
Thus, the correct answer is:
D. [tex]\( ME = KE + GPE \)[/tex]
