Answer:
16.58 m/s
Explanation:
Let g = 10 m/s2
The total (kinetic and potential) energy of Gayle at top of the hill (15 m) is
[tex]E_g = E_k + E_p = m_gv_g^2/2 + gm_gh_g[/tex]
where [tex]m_g[/tex] = 50 + 5 = 55 kg is the mass of Gayle and the sledge, [tex]v_g[/tex] = 4m/s is the speed of Gayle on top, [tex]h_g[/tex] = 15 m is the vertical height of Gayle on top.
[tex]E_g = 55*4^2/2 + 10*55*15 = 8690 J[/tex]
Similarly, the potential energy of her brother at 15 - 5 = 10m high is:
[tex]E_b = E_p = m_bgh_b[/tex]
where [tex]m_b[/tex] = 30kg is the mass of the brother, [tex]h_b[/tex] = 10 m is the vertical height of the brother when he jumps on.
[tex]E_b = 30*10*10 = 3000 J[/tex]
The total energy is:
[tex] E = E_g + E_b = 8690 + 3000 = 11690 J[/tex]
According to energy conservation, all of this is converted to kinetic energy at the bottom:
[tex] E_k = 11690 J[/tex]
[tex]MV^2/2 = 11690 J[/tex]
where M = 50 + 30 + 5 = 85 kg is the total system mass. V is the system mass at the bottom
[tex]V^2 = 2*11690/M = 2*11690/85 = 275[/tex]
[tex]V = \sqrt{275} = 16.58 m/s[/tex]
