Bang bang, you shot me down...

Classical Mechanics Level 5

A man sitting in a hot air balloon floating \(h\) meters above the ground drops an object towards the ground, and at the same time fires a gunshot.

An observer on the ground, standing on the ground right next to the place of the impact of the object, measures a time difference \(\Delta t = 3s\) between the arrival of the sound of the shot and the impact of the object.

What is the sum of the two possible heights \(h_{1}\) and \(h_{2}\), if both heights are rounded down to the nearest lower integer?

\(\textbf{Details and assumptions}\)

  • The temperature of the air is \(T = 14.5°C\)
  • The ideal gas constant is \(R = 8.31 \frac {J}{mol \cdot K}\)
  • The molecular mass of air is \(MM = 28.96 \frac {g}{mol}\)
  • The adiabatic index of air is \(1.4\)
  • The gravitational acceleration is \(g = 9.8 \frac {m}{s^2}\)
  • For simplicity, round off the speed of sound \(v_{s}\) to the nearest integer
  • Both heights \(h_{1}\) and \(h_{2}\) are rounded down to the nearest lower integer before being added together
  • Assume no air drag acts upon the object
Image credit: Wikipedia Tommaso.gavioli

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