Let's solve the problem step-by-step.
We need to identify the correct equation that relates kinetic energy ([tex]\( KE \)[/tex]), mass ([tex]\( m \)[/tex]), and velocity ([tex]\( v \)[/tex]).
The formula for kinetic energy is derived from the principles of physics, specifically from the work-energy theorem. For an object with mass [tex]\( m \)[/tex] moving with velocity [tex]\( v \)[/tex], the kinetic energy is mathematically given by:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
Let's analyze the given options:
A. [tex]\( KE = \frac{1}{2} m^2 v \)[/tex]
Here, the mass [tex]\( m \)[/tex] is squared and the velocity [tex]\( v \)[/tex] is to the first power. This form is incorrect because kinetic energy does not involve mass squared, and velocity should be squared.
B. [tex]\( KE = \frac{1}{2} m v^2 \)[/tex]
This matches our earlier stated formula for kinetic energy. The mass [tex]\( m \)[/tex] and the square of velocity [tex]\( v \)[/tex] are correctly included. Therefore, this looks like the correct option.
C. [tex]\( KE = \frac{1}{2} m v \)[/tex]
In this option, both mass [tex]\( m \)[/tex] and velocity [tex]\( v \)[/tex] are to the first power. This does not correctly represent kinetic energy because velocity should be squared, not to the first power.
D. [tex]\( KE = \frac{1}{2} m v^3 \)[/tex]
Here, the velocity [tex]\( v \)[/tex] is cubed. This is incorrect because kinetic energy involves the square of velocity, not the cube.
After reviewing all choices, the correct relationship is:
[tex]\[ \boxed{KE = \frac{1}{2} m v^2} \][/tex]
Therefore, the correct answer is:
B. [tex]\( KE = \frac{1}{2} m v^2 \)[/tex]
