When, the rock is located at the maximum height, there is no kinetic energy and the total mechanical energy is the potential energy:
[tex]E_1=U_1=mgh_{max}[/tex]When the rock is located at a height such that the kinetic energy is twice the potential energy, we have:
[tex]\begin{gathered} E_2=U_2+K_2 \\ \\ =U_2+2U_2 \\ \\ =3U_2 \\ \\ =3mgh \end{gathered}[/tex]Since the mechanical energy is conserved, then:
[tex]\begin{gathered} 3mgh=mgh_{max} \\ \\ \Rightarrow h=\frac{1}{3}h_{max} \end{gathered}[/tex]Replace h_max=21.0m to find the height at which the kinetic energy of the rock is twice the potential energy of the rock:
[tex]h=\frac{1}{3}(21.0m)=7.0m[/tex]Therefore, the answer is: 7m.
