We apply the conservation of momentum principle.
M V + m v = 0 where V is the velocity of the ballon
R H S is 0 as both of them were stationary initially.
V = - m v / M. V is in the direction opposite to v of the passenger.
As passenger is sliding down, balloon climbs up.
When the passenger stops sliding , he/she exerts a force on the balloon to stop himself/herself. That force will stop the balloon too. With the force balloon exerts on him, he will stop sliding.
Energy is conserved as, passenger loses some height and the balloon climbs up gaining potential energy, The kinetic energy comes from the potential energy lost by person.
as MV = - mv
To prove Mg H + 1/2 M V² = mg h + 1/2m v²
H = height climbed by baloon in time t = V * t
h = height lost by person in time t = v * t = MV/m t
So to prove Mg V t + 1/2 M V² = m g (MV/m) t + 1/2 m v²
= Mg V t + 1/2 M²V² / m as - mv = MV