To solve the given question, we must determine which of the provided options correctly applies the work-energy theorem. According to this theorem, the work done on an object is equal to the change in its kinetic energy.
The kinetic energy (KE) of an object with mass [tex]\( m \)[/tex] and velocity [tex]\( v \)[/tex] is given by the formula:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
The work done [tex]\( W \)[/tex] on the object can be described as the change in kinetic energy:
[tex]\[ W = \Delta KE = KE_{final} - KE_{initial} \][/tex]
Where [tex]\( KE_{final} \)[/tex] is the kinetic energy with the final velocity [tex]\( v_f \)[/tex], and [tex]\( KE_{initial} \)[/tex] is the kinetic energy with the initial velocity [tex]\( v_i \)[/tex].
[tex]\[ KE_{final} = \frac{1}{2} m v_f^2 \][/tex]
[tex]\[ KE_{initial} = \frac{1}{2} m v_i^2 \][/tex]
Therefore, the change in kinetic energy [tex]\( \Delta KE \)[/tex] is:
[tex]\[ \Delta KE = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 \][/tex]
[tex]\[
\Delta KE = \frac{1}{2} m (v_f^2 - v_i^2)
\][/tex]
Thus, the formula for the work done [tex]\( W \)[/tex] using the work-energy theorem is:
[tex]\[ W = \frac{1}{2} m (v_f^2 - v_i^2) \][/tex]
Comparing this with the given options, the correct choice is:
[tex]\[ W = \Delta KE = \frac{1}{2} m (v_f^2 - v_i^2) \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
