Sure! Let's find the correct equation that represents the mechanical energy (ME) of a system. Mechanical energy is the sum of kinetic energy (KE) and potential energy (PE).
1. Kinetic Energy (KE) is given by the formula:
[tex]\[
KE = \frac{1}{2}mv^2
\][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the object,
- [tex]\( v \)[/tex] is the velocity of the object.
2. Potential Energy (PE), specifically gravitational potential energy, is given by the formula:
[tex]\[
PE = mgh
\][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the object,
- [tex]\( g \)[/tex] is the gravitational acceleration (approximately [tex]\( 9.8 \, \text{m/s}^2 \)[/tex] on Earth),
- [tex]\( h \)[/tex] is the height of the object above a reference point.
Now, the total mechanical energy (ME) is the sum of kinetic energy and potential energy:
[tex]\[
ME = KE + PE
\][/tex]
Substituting the formulas for kinetic and potential energy, we get:
[tex]\[
ME = \frac{1}{2} mv^2 + mgh
\][/tex]
Now, let's look at the options provided:
- Option A: [tex]\( ME = \frac{1}{2} mv^2 \times mgh \)[/tex]
- This option incorrectly multiplies kinetic and potential energy terms.
- Option B: [tex]\( ME = \frac{1}{2} mv^2 - mgh \)[/tex]
- This option incorrectly subtracts potential energy from kinetic energy.
- Option C: [tex]\( ME = \frac{1}{2} mv^2 + mgh \)[/tex]
- This option correctly adds kinetic and potential energy.
- Option D: [tex]\( ME = \frac{\frac{1}{2} mv^2}{mgh} \)[/tex]
- This option incorrectly divides kinetic energy by potential energy.
From the above explanations, the correct equation is:
[tex]\[
ME = \frac{1}{2} mv^2 + mgh
\][/tex]
Thus, the correct answer is Option C.
