by parts)}\\ &=\gamma m_0 U^2-m_0\int_0^U\frac{udu}{\sqrt{1-\frac{u^2}{c^2}}}\\ &=\gamma m_0 U^2-m_0\Big[ c^2\sqrt{1-\frac{u^2}{c^2}} \Big]_0^U\\ &=\gamma m_0 U^2-m_0c^2+m_0c^2\sqrt{1-\frac{U^2}{c^2}}...by parts)}\\ &=\gamma m_0 U^2-m_0\int_0^U\frac{udu}{\sqrt{1-\frac{u^2}{c^2}}}\\ &=\gamma m_0 U^2-m_0\Big[ c^2\sqrt{1-\frac{u^2}{c^2}} \Big]_0^U\\ &=\gamma m_0 U^2-m_0c^2+m_0c^2\sqrt{1-\frac{U^2}{c^2}}\\ &=\gamma \left(m_0 U^2+m_0c^2\left(1-\frac{U^2}{c^2}\right)\right)-m_0c^2\\ &=m_0c^2(\gamma -1) \end{aligned}\] Since the object started at rest (with a speed \(u=0\)) the above integral corresponds to what we would call the kinetic energy of the object, with a speed, \(u\): \[\begin{aligned} K=…
