The change in the square of the velocity.
Explanation
The problem is based on the Work-Energy Theorem, which states that the net work done on an object is equal to the change in its kinetic energy. Kinetic energy is a function of the mass of the body and the square of its velocity.
Analysis
According to the Work-Energy Theorem:
Work Done ($W$) = Change in Kinetic Energy ($\Delta KE$)
The formula for kinetic energy is:
$KE = \frac{1}{2}mv^2$
Where:
- $m$ is the mass of the object.
- $v$ is the velocity of the object.
When the speed increases from an initial velocity ($v_i$) to a final velocity ($v_f$), the work done is calculated as:
$W = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$
$W = \frac{1}{2}m(v_f^2 - v_i^2)$
Since the mass ($m$) is constant, the work done is directly proportional to the term $(v_f^2 - v_i^2)$, which represents the change in the square of the velocity.
Therefore, the work done is not proportional to the simple change in velocity ($v_f - v_i$) or the sum of velocities, but specifically to the difference of their squares.
Key Takeaway: Under the Work-Energy Theorem, work done is defined as the change in kinetic energy. Since kinetic energy depends on the square of velocity ($v^2$), work done is proportional to the change in the square of the velocity.