Final answer:
Mechanical energy in a one-dimensional particle motion with known potential energy is conserved, meaning it remains constant over time. The mechanical energy is the sum of kinetic and potential energy, which can be calculated using position-time relationship in systems with conservative forces.
Step-by-step explanation:
Writing mechanical energy as a function of time involves understanding that in a system where this energy is conserved, the total energy remains constant over time. In a one-dimensional particle motion with a known potential energy function, the mechanical energy is the sum of kinetic energy (KE) and potential energy (PE), and can be expressed as E = KE + PE. For cases where we neglect non-conservative forces such as air resistance, and potential energy can be derived from height (such as in a gravitational field), we can use the formula U(y) = mgy where m represents mass, g the acceleration due to gravity, and y the height.
By also considering the particle's initial conditions and applying the conservation of energy principle, we can substitute known expressions of potential energy as functions of position into E = KE + U and integrate to find the particle's position as a function of time. With the position known as a function of time, we can then derive other quantities like velocity and hence kinetic energy as functions of time. Therefore, mechanical energy as a function of time, in a conservative system, will be constant and equal to the initial total energy of the system.