Answer
The second ledge should be placed at a height of $1.27~meters$ above the ground.
Work Step by Step
We can use conservation of energy to find Jones' speed $v_0$ just before the collision:
$KE = U_g$
$\frac{1}{2}m_0v_0^2 = m_0gh$
$v_0 = \sqrt{2gh}$
$v_0 = \sqrt{(2)(9.80~m/s^2)(3.70~m})$
$v_0 = 8.516~m/s$
We can use conservation of momentum to find their speed $v_f$ just after the collision:
$m_f~v_f= m_0~v_0$
$v_f= \frac{m_0~v_0}{m_f}$
$v_f= \frac{(78.0~kg)~(8.516~m/s)}{78.0~kg+55.0~kg}$
$v_f = 4.99~m/s$
We can use conservation of energy to find the height that they reach:
$U_g = KE$
$m_fgh = \frac{1}{2}m_fv_f^2$
$h = \frac{(v_f)^2}{2g}$
$h = \frac{(4.99~m/s)^2}{(2)(9.80~m/s^2)}$
$h = 1.27~m$
The second ledge should be placed at a height of $~1.27~meters~$ above the ground.
