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# 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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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)

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
$$\displaystyle{h}={\frac{{{v}}}{{{g}}}}{\left({v}+{g}{T}\pm\sqrt{{{v}^{{2}}+{2}{g}{v}{T}}}\right)}$$