explain (the application of density in the operation of hot air balloons, and submarines)?

Answer 1

An object that is more dense than surroundings will sink, an object that is less dense than surroundings will float.

Thus, in a hot air balloon (an open system), the heat causes the gas to expand, and as it is not confined, the density drops and the balloon floats or rises. As it cools, the density rises and thus the balloon drops or lowers.

In a submarine (a closed system), water or ballast is pumped in pressurising the air in the tanks, increasing the density causing the submarine to sink or submerge. Then the pressure is released, the density drops and the submarine rises.

Answer 2

You probably know that an atmospheric Thermal is air rising due to a decrease in density when the air is heated.
Both the Hot-air balloon and the Submarine operate on the principles density change.

1...The Hot air balloon uses this principle to operate by trapping a man-made thermal in the material of the balloon.
The thermal is a powerful one as it's formed by burning propane which causes the much lower density air to rise into and inflate the balloon causing it to lift of the ground.
2...A submarine works in the same way by changing the vessels' density. It is constructed with air filled ballast (buoyancy) tanks which keeps it afloat. (Lower density than water).
When it dives, the air in the ballast tanks is replaced with water and its density increases causing it to sink.
Its depth can be controlled by controlling the air to water ratio in the ballast tanks. When ready to re-surface, the water in the ballast tanks is 'blown' out with compressed air.