Final answer:
The statement is false; according to the conservation of mechanical energy principle, the sum of initial kinetic and potential energy equals the sum of final kinetic and potential energy in the absence of non-conservative forces. Changes in mechanical energy are due to work being done on or by the system, so without work there should be no change if only conservative forces are involved.
Step-by-step explanation:
The statement that 'the kinetic energy before plus the kinetic energy after equals the potential energy before plus the potential energy after' is false. Instead, the conservation of mechanical energy principle states that in the absence of non-conservative forces like friction, the total mechanical energy of a system remains constant. This means KEi + PEi = KEf + PEf, where i denotes initial states and f denotes final states. The equation KEi + PEi = KEf + PEf is a form of the work-energy theorem for conservative forces, illustrating mechanical energy conservation.
Is it possible for the sum of kinetic energy and potential energy of an object to change without work having been done on the object? The answer is typically no, as changes in mechanical energy are generally the result of work being done on or by the system unless there is a force acting on the object which does not conserve mechanical energy, such as friction or air resistance. When an external force does work on a system, energy is transferred into or out of the system. If no work is done, the total mechanical energy (the sum of kinetic and potential energy) of a system that is subjected only to conservative forces should be conserved.