Question

A physics student with too much free time drops a watermelon fromthe roof of a building. He hears the sound of the watermelon going"splat" after a tim

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asked 2021-03-18
A physics student with too much free time drops a watermelon fromthe roof of a building. He hears the sound of the watermelon going"splat" after a time interval of \(\displaystyle\triangle{t}\).
You may ignore air resistance. How high isthe building? (The speed of sound is \(\displaystyle{v}_{{s}}\). Take the free fall acceleration to be g)

Answers (1)

2021-03-19
I'll use slightly different notation to avoid using subscripts
\(\displaystyle{T}=\triangle{t}\)
\(\displaystyle{v}={v}_{{s}}\)
The time it takes to hit the ground:
\(\displaystyle{0}={h}-{\frac{{{{>}_{{1}}^{{2}}}}}{{{2}}}}\)
\(\displaystyle{t}_{{1}}=\sqrt{{{\frac{{{2}{h}}}{{{g}}}}}}\)
The time for the sound to tavel
\(\displaystyle{h}={v}{t}_{{2}}\)
\(\displaystyle{t}_{{2}}={\frac{{{h}}}{{{v}}}}\)
\(\displaystyle{T}={t}_{{1}}+{t}_{{2}}=\sqrt{{\frac{{{2}{h}}}{{{g}}}}}+{\frac{{{h}}}{{{v}}}}\)
\(\displaystyle{\frac{{{2}{h}}}{{{g}}}}={\left({T}-{\frac{{{h}}}{{{v}}}}\right)}^{{2}}={T}^{{2}}-{\frac{{{2}{T}}}{{{v}}}}{h}+{\frac{{{h}^{{2}}}}{{{v}^{{2}}}}}\)
This simplifies to
\(\displaystyle{g}{h}^{{2}}-{2}{v}{\left({v}+{g}{T}\right)}{h}+{g}{v}^{{2}}{T}^{{2}}={0}\)
Using the quadratic formula and simplifying
\(\displaystyle{h}={\frac{{{v}}}{{{g}}}}{\left({v}+{g}{T}\pm\sqrt{{{v}^{{2}}+{2}{g}{v}{T}}}\right)}\)
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