Hot Air Balloons
One of the earliest forms of flight, hot air balloons demonstrate how beautiful and simple Newton's 2^{nd} law and the principles of buoyancy can be.
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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 life up in the air, the net force on it must be point upward. Using Newton's 2^{nd} law
\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 that is pointing vertically downward. The buoyant force, from Archimedes principle is
\begin{align*}F_{up}=\rho Vg\end{align*}
where \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 up by the burners, it is less dense than the surrounding cooler air outside the balloon. When enough lift is generated the buoyant force is greater than the weight of the balloon and occupants and the balloon begins to accelerate upwards.
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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?