A rifle bullet passes through two vertical sheets of paper d…

A rifle bullet passes through two vertical sheets of paper $d = 20\ \mathrm{m}$ apart. The hole in the second sheet is $h = 2\ \mathrm{cm}$ lower than the hole in the first. The bullet was moving horizontally at the first sheet. Find its speed.

Explanation of Method

Speed of a bullet from the drop between two sheets.

Step 1: Time between the sheets

Starting horizontally, the bullet falls freely by $h$ in the time it needs to cross $d$: \[h = \tfrac12 g t^2 \;\Longrightarrow\; t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{0.04}{9.81}} \approx 0.0639\ \mathrm{s}.\]

Step 2: Speed

\[ v = \frac{d}{t} = d\sqrt{\frac{g}{2h}} = \frac{20}{0.06386}. \]

\[v \approx 313\ \mathrm{m/s}\]

$d = 20\ \mathrm{m}$ $h = 2\ \mathrm{cm}$ $\text{sheet 1}$ $\text{sheet 2}$ $v$

Not to scale: the 2 cm drop is hugely exaggerated. Between the sheets the bullet falls freely for the time it takes to cover 20 m horizontally.

Result

\[\boxed{\,v \approx 313\ \mathrm{m/s}\,}\]

Remark

Note. Air resistance, which would slow a real bullet slightly over 20 m, is ignored. The original value 313.5 m/s comes from rounding $t$ to 0.0638 s.