Vertical motion of a balloon in air

The equations of motion of a balloon ascending in air are formed taking into account exponential law of the air density and the air as viscous medium and are completely solved. The balloon rises, comes to rest and begins to float at a certain height. It is established that if certain conditions as regards the weight and size of the balloon are fulfilled, then it first executes a damped oscillatory motion about such a height as the mean position of rest with a small amplitude gradually diminishing, before coming to rest at that height. If these conditions are not satisfied, the balloon straight way rises to this height and thereafter begins to float. Two numerical examples are illustrated