Energy concepts are fundamentally important for describing and analyzing systems ranging from subatomic particles to spiral galaxies. In general, students first encounter such concepts in introductory courses that typically focus on forms of energy, energy transfer, and conservation laws. Within these courses, conservation of mechanical energy is particularly useful for solving problems (since the physical quantities being analyzed do not require vector analysis). However, strictly speaking, conservation of mechanical energy is valid only when no energy is transferred across the boundary of the system being considered.