Related Rates

When two or more physical quantities are linked by a single relationship, a change in one variable necessarily affects the others. This interdependence forms the basis of related rates analysis, which examines how different quantities change with respect to time. A classic physical example is an expanding balloon, where the size of the balloon changes continuously as air is added.

For a hot air balloon, the inflated envelope is commonly idealized as a perfect sphere to simplify mathematical analysis. Under this assumption, the volume of the balloon depends directly on its radius according to the geometric relationship

As air is pumped into the balloon, the volume increases steadily over time, and this increase forces the radius to expand as well. Differentiating this relationship with respect to time links the rate at which air enters the balloon to the rate at which the radius changes.

This connection is essential for ensuring safe inflation. Air enters the balloon at a constant rate, producing a known rate of volume increase. At a particular instant, such as when the radius is 2 meters and the inflow rate is 0.5 cubic meters per second, these known values can be used to determine how rapidly the radius is expanding at that moment. This rate represents how quickly the balloon fabric is being stretched.

Material limitations make this calculation critical. The balloon envelope can withstand only a maximum expansion rate before tearing occurs. Importantly, the rate of radial expansion is greatest when the balloon is small and decreases as the balloon grows. By evaluating the expansion rate during the early stages of inflation, operators can ensure that rapid initial growth does not exceed the material’s rupture threshold, as a result of maintaining structural integrity and safety.