Verify that in Example 9.6 the momentum transferred to the ball by the gravitational force is \(\Delta p_{x}=m_{\mathrm{b}}\left(v_{x, \mathrm{f}}-v_{x, \mathrm{i}}\right)\).

Data from Example 9.6

A ball of inertia \(m_{\mathrm{b}}\) is released from rest and falls vertically. What is the ball's final kinetic energy after a displacement \(\Delta x=x_{\mathrm{f}}-x_{\mathrm{i}}\) ?