### Hot Air Balloons

One of the earliest forms of flight, hot air balloons demonstrate how Newton's 2^{nd} law and the principles of buoyancy can work together.

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- Newton's 2
^{nd}law states that the sum of all the forces acting on an object is equal to the mass of that object times its acceleration. Therefore, for a hot air balloon to ascend, the net force on it must be point upward:

\begin{align*}F_{up}-F_g=ma\end{align*}

- \begin{align*}F_{up}\end{align*} is the buoyant force directed upward, and \begin{align*}F_g\end{align*} is the force due to gravity, which points downward. The buoyant force, from Archimedes principle is shown in the following equation:

\begin{align*}F_{up}=\rho Vg\end{align*}

\begin{align*}\rho\end{align*} is the density, \begin{align*}V\end{align*} is the displaced volume, and \begin{align*}g\end{align*} is gravity.

- Since the air inside the hot air balloon is heated by burners, it is less dense than the cooler air outside the balloon. When enough lift is generated, the buoyant force is greater than the weight of the balloon and the balloon can begin to accelerate upwards.

#### Explore More

Using the information provided above, answer the following questions.

- Show mathematically what Newton's 2
^{nd}law says when the balloon's burners are heating up the air inside the balloon but there is no upward lift. - What is the relationship between temperature and density?
- If the density of the air inside the balloon was greater than the density outside of the balloon, would lift be generated?