Question

A weather hot-air balloon uses the low density of hot air to generate a buoyant force from the atmospheric pressure. If the balloon is accelerating upwards at a rate of 1.5 m/s², what is the ratio between the density of the hot air in the balloon and the cold air outside (two decimal points)?

Answer 1

The balloon experiences 2 vertical forces:

1. Gravitational force = mg = dvg downwards, where m is mass of balloon, g is gravitational acceleration, d is density of hot air inside the balloon and v is volume of the balloon.

2. Buoyant force = Dvg upwards, where D is density of cold air outside , v is volume of balloon and g is gravitational acceleration.

Now, net force = mass*acceleration.

Here,acceleration = 1.5 m/s2 upwards.

So, Dvg - dvg = m*1.5, where m is mass of balloon.

Also, m = dv, where d is density of hot air inside the balloon and v is volume of the balloon.

So,Dvg - dvg = dv*1.5

=>Dg-dg = 1.5 d

=> Dg = (1.5+g)d

=>d/D = g/(1.5+g) = 9.8/(9.8+1.5) = 0.87

So, required ratio = 0.87