A balloon of volume $$V,$$ contains a gas whose density is to that of the air at the earth’s surface as $$1:15.$$  If the envelope of the balloon be of weight $$w$$ but of negligible volume, find the acceleration with which it will begin to ascend.

Solution :
Let $$\sigma $$ be the density of the gas, then that of the air is $$15\sigma .$$  Then the weight of the balloon = weight of the gas + weight of the envelope $$ = Vg\sigma + w.$$
If $$f$$ be the required acceleration of the balloon acting vertically upward and then from "mass $$ \times $$ acceleration = forces acting in the sense of acceleration" we get $$\frac{{\left( {Vg\sigma + w} \right)}}{g} \times a = $$     force of buoyance - wt. of the balloon with gas $$ = V15\sigma g - \left( {Vg\sigma + w} \right)$$
$${\text{or}}\,\,a = \left( {\frac{{14Vg\sigma - w}}{{Vg\sigma + w}}} \right) \times g$$