Sure, let's go through the concepts of mechanical energy, kinetic energy, and potential energy step by step to identify the correct equation.
1. Mechanical Energy (ME):
Mechanical energy is the sum of kinetic and potential energies in a system. It represents the total energy of the system due to both its motion and its position.
2. Kinetic Energy (KE):
Kinetic energy is the energy that an object possesses due to its motion. For an object of mass [tex]\( m \)[/tex] moving at velocity [tex]\( v \)[/tex], kinetic energy is given by:
[tex]\[
KE = \frac{1}{2} m v^2
\][/tex]
3. Potential Energy (PE):
Potential energy is the energy stored in an object due to its position in a force field, such as gravity. For example, the gravitational potential energy of an object of mass [tex]\( m \)[/tex] at height [tex]\( h \)[/tex] in a gravitational field with acceleration [tex]\( g \)[/tex] is:
[tex]\[
PE = mgh
\][/tex]
4. Relationship between ME, KE, and PE:
The mechanical energy of a system is the sum of its kinetic energy and potential energy. This relationship can be written as:
[tex]\[
ME = KE + PE
\][/tex]
Given the options, this explains why the correct equation is:
[tex]\[
ME = KE + PE
\][/tex]
Thus, the correct equation showing the relationship between mechanical energy, kinetic energy, and potential energy is:
[tex]\[
ME = KE + PE
\][/tex]
