# Conservation of energy question

Lets say you have a helium balloon or hot air balloon, carrying a weight of mass $m$. When you release the balloon, it ascends, and at any point if it is at altitude $h$, the potential energy in the payload is $mgh$. As the balloon rises, this increases. But where is this energy coming from?

It seems that if you look at the big picture, you can view the situation as the air "falling", rather than the balloon rising. But still, if you drop the payload, it's going to come crashing down with a lot of energy. Where did this energy come from? Certainly the person who filled the balloon didn't expend that much energy.

• The answer is in your question.... The balloon wouldn't be able to fly if you don't have that device which produces a flame. The heat radiated by the flaming device allows to balloon to rise. Commented Aug 5, 2017 at 10:38
• I don't understand your second paragraph. What is the payload? What is crashing? Commented Aug 5, 2017 at 11:37
• OK, forget about hot air balloons. Imagine you have a large helium balloon attached by a string to a rock. Suppose it goes up 10000 meters and then the string tears and rock falls. At that point the rock has a lot of energy. Where did it come from? Commented Aug 5, 2017 at 11:45
• Helium is lighter than air(that's why it goes up through buoyancy) and hence the work done to collect all that helium and compress it will be enough for the kinetic energy of the rock. Commented Aug 5, 2017 at 15:17

On the ground the total mass of the balloon (and contents) is $m$ and its weight is $mg$.
When the balloon rise a vertical height $h$ the balloon gains gravitational potential energy $mgh$.

Now for the balloon to rise it must displace a volume of air which has a weight equal to (or very slightly more than) the weight of the balloon $mg$ - Archimedes.

When the balloon rises to a height $h$ a volume of air which has a weight equal to the weight of the balloon "falls" a height $h$ - a volume of air which is displaced by the balloon when at a height $h$ now occupies the volume vacated by the balloon when it was on the ground.
So the loss in gravitational potential of the displaced air is $mgh$ which is equal to the gravitational potential energy gained by the balloon.

• But the volume of air at height $h$ has less mass than $m$, since density decreases with height. Commented Aug 7, 2017 at 1:37
• @stafusa Does not the volume of balloon increase as it goes higher? Commented Aug 7, 2017 at 4:57
• No, a hot air balloon usually doesn't and for a helium one it depends on the elasticity of its material. Commented Aug 7, 2017 at 9:26