Sure, let's explore each option to identify the correct relationship between mechanical energy (ME), kinetic energy (KE), and potential energy (PE).
1. [tex]\( ME = KE + PE \)[/tex]
- This equation suggests that the mechanical energy is the sum of kinetic energy and potential energy.
- Typically, this is the correct and well-known relationship in physics. Mechanical energy in a closed system without non-conservative forces is conserved and is the sum of kinetic and potential energy.
2. [tex]\( ME + KE = PE \)[/tex]
- This would imply that the sum of mechanical energy and kinetic energy equals potential energy.
- This is not a standard formulation in physics and does not align with the conservation laws of mechanical energy.
3. [tex]\( ME = KE - PE \)[/tex]
- This suggests that mechanical energy is the difference between kinetic energy and potential energy.
- This again does not conform to the typical definitions and relationships used in physics.
4. [tex]\( ME + PE = KE \)[/tex]
- This would imply that the sum of mechanical energy and potential energy equals kinetic energy.
- This is also unconventional and incorrect according to the principle of conservation of mechanical energy.
Given our exploration of these options, the correct relationship is:
[tex]\[ ME = KE + PE \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{0} \][/tex]
