A 4.92E+2 kg hot-air balloon takes off from rest at the surface of the earth. The nonconservative wind and lift forces take the balloon up, doing +9.81E+4 J of work on the balloon in the process. At what height above the surface of the earth does the balloon have a speed of 7.90 m/s?

Answer 1

Do an energy balance.

Work done on balloon = (increases in KE) + (increase in gravitational potential energy)

Solve for the G.P.E change. You can use that number to compute the altitude.

It is OK to assume g is constant over the altitude range of a balloon.