To determine which equation correctly represents the mechanical energy of a system, we need to understand the fundamental components of mechanical energy.
Mechanical energy is the total energy of a system that is available to do work. It is the sum of two types of energy: kinetic energy (KE) and potential energy (PE).
1. Kinetic Energy (KE): This is the energy that an object possesses due to its motion. It can be represented by the formula:
[tex]\[
\text{KE} = \frac{1}{2} m v^2
\][/tex]
where [tex]\( m \)[/tex] is the mass of the object, and [tex]\( v \)[/tex] is its velocity.
2. Potential Energy (PE): This is the energy that an object possesses due to its position in a gravitational field. It can be represented by the formula:
[tex]\[
\text{PE} = mgh
\][/tex]
where [tex]\( m \)[/tex] is the mass of the object, [tex]\( g \)[/tex] is the acceleration due to gravity, and [tex]\( h \)[/tex] is the height above the reference point.
The total mechanical energy (ME) of the system is the sum of the kinetic energy and the potential energy:
[tex]\[
\text{ME} = \text{KE} + \text{PE}
\][/tex]
Substituting the formulas for KE and PE, we get:
[tex]\[
\text{ME} = \frac{1}{2} m v^2 + mgh
\][/tex]
Therefore, the correct representation for the mechanical energy (ME) of a system is:
[tex]\[
\text{ME} = \frac{1}{2} m v^2 + mgh
\][/tex]
Comparing this with the given options:
- Option A: [tex]\( ME = \frac{1}{2} m v^2 \times m g h \)[/tex]
- This is incorrect as it multiplies the kinetic energy and potential energy.
- Option B: [tex]\( ME = \frac{\frac{1}{2} m v^2}{m g h} \)[/tex]
- This is incorrect as it divides the kinetic energy by the potential energy.
- Option C: [tex]\( ME = \frac{1}{2} m v^2 - m g h \)[/tex]
- This is incorrect as it subtracts the potential energy from the kinetic energy.
- Option D: [tex]\( ME = \frac{1}{2} m v^2 + m g h \)[/tex]
- This is correct. This correctly adds the kinetic energy and potential energy to give the total mechanical energy.
Hence, the correct answer is:
Option D: [tex]\( ME = \frac{1}{2} m v^2 + m g h \)[/tex]
