To determine which equation correctly represents the law of conservation of energy in a closed system, let's start by understanding the fundamental principle underlying this law.
The law of conservation of energy states that within a closed system, the total energy remains constant over time. This means that the energy can neither be created nor destroyed but can only transform from one form to another. The primary forms of mechanical energy in this context are kinetic energy (KE) and potential energy (PE).
Given this, the total initial energy (which includes initial kinetic energy [tex]\(KE_i\)[/tex] and initial potential energy [tex]\(PE_i\)[/tex]) should equal the total final energy (which includes final kinetic energy [tex]\(KE_f\)[/tex] and final potential energy [tex]\(PE_f\)[/tex]).
Let's examine each provided option:
1. [tex]\( K E_i + P E_i = K E_{f} + P E_{f} \)[/tex]
This equation states that the sum of the initial kinetic energy and initial potential energy is equal to the sum of the final kinetic energy and final potential energy. This matches the law of conservation of energy, as it demonstrates that the total mechanical energy is conserved.
2. [tex]\( P E_i + P E_f = K E_i + K E_f \)[/tex]
This equation incorrectly swaps the sum of initial and final energies between potential and kinetic forms, which does not represent the law of conservation of energy.
3. [tex]\( K E_i - K E_f = P E_i - P E_f \)[/tex]
This equation indicates that the difference in kinetic energies is equal to the difference in potential energies, which does not directly represent the conservation of total energy.
4. [tex]\( K E_i - P E_f = P E_i - K E_f \)[/tex]
This equation mixes kinetic and potential energies inappropriately, which also does not reflect the law of conservation of energy.
After evaluating all the options, we see that the first equation correctly represents the law of conservation of energy.
Therefore, the correct equation is:
[tex]\[ K E_i + P E_i = K E_{ f } + P E_{ f } \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
