Work and Energy
Beginner
work-energy
Source: High school physics (Chinese)
Problem
A ball of mass $m$ is thrown vertically downward from a height $h$ with an initial speed $v_i$. It reaches the ground with a final speed $v_f$.
Find the work $W_{air}$ done by the ball to overcome air resistance during its fall.
$$W_{air} = mgh + \frac{1}{2}m(v_i^2 - v_f^2)$$
The work done against a non-conservative force like air resistance equals the decrease in the total mechanical energy of the system. The initial mechanical energy (at height $h$) is the sum of its kinetic and potential energies, $E_i = K_i + U_i = \frac{1}{2}mv_i^2 + mgh$. The final mechanical energy at ground level ($h=0$) is purely kinetic, $E_f = K_f = \frac{1}{2}mv_f^2$. The work done against resistance is the energy dissipated:
$$W_{air} = E_i - E_f = \left(\frac{1}{2}mv_i^2 + mgh\right) - \frac{1}{2}mv_f^2$$ $$W_{air} = mgh + \frac{1}{2}m(v_i^2 - v_f^2)$$