To determine which equation correctly relates kinetic energy ([tex]\(KE\)[/tex]), mass ([tex]\(m\)[/tex]), and velocity ([tex]\(v\)[/tex]), let's recall the basic principles of kinetic energy.
Kinetic energy is the energy an object possesses due to its motion. The standard formula for kinetic energy in classical mechanics is given by:
[tex]\[ KE = \frac{1}{2} mv^2 \][/tex]
Here is the reasoning for each option:
- Option A:
[tex]\[ KE = \frac{1}{2} m^2 v \][/tex]
This suggests that kinetic energy is proportional to the square of the mass [tex]\(m\)[/tex] and linearly proportional to the velocity [tex]\(v\)[/tex]. This is not correct based on the standard formula for kinetic energy.
- Option B:
[tex]\[ KE = \frac{1}{2} mv^2 \][/tex]
This matches the standard and well-known formula for kinetic energy. It correctly indicates that kinetic energy is proportional to the mass [tex]\(m\)[/tex] and the square of the velocity [tex]\(v\)[/tex].
- Option C:
[tex]\[ KE = \frac{1}{2} mv \][/tex]
This suggests that kinetic energy is linearly proportional to both the mass [tex]\(m\)[/tex] and the velocity [tex]\(v\)[/tex]. This does not match the standard kinetic energy formula.
- Option D:
[tex]\[ KE = \frac{1}{2} mv^3 \][/tex]
This suggests that kinetic energy is proportional to the mass [tex]\(m\)[/tex] and the cube of the velocity [tex]\(v\)[/tex]. This is not correct based on the kinetic energy formula.
After carefully evaluating the options, it is evident that option B, [tex]\( KE = \frac{1}{2} mv^2 \)[/tex], is the correct relationship between kinetic energy, mass, and velocity.
Therefore, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
or, explicitly, option B:
[tex]\[ KE = \frac{1}{2} mv^2 \][/tex]
