fertilizeki

2022-01-12

While anchored in the middle of a lake, you count exactly three waves hitting your boat every 10 s. You raise anchor and start motoring slowly in the same direction the waves are going. When traveling at 1.5 m/s, you notice that exactly two waves are hitting the boat from behind every 10 s. What is the speed of the waves on the lake?

Melissa Moore

Step 1
Solve for ${f}_{s}$:
${f}_{s}=\frac{3.0}{10}s=0.30Hz$
When you are moving, the measured frequency is expressed as:
${f}_{-}=\frac{2.0}{10}s=0.20Hz$
By application of the equation ${f}_{-}=\left(1\frac{{v}_{s}}{v}\right){f}_{s}$, we have:
$\frac{{f}_{-}}{{f}_{+}}=1-\frac{{v}_{s}}{v}$
$\frac{{v}_{s}}{v}=1-\frac{{f}_{-}}{{f}_{s}}$
$v={v}_{s}\left(\frac{1}{1-\frac{{f}_{-}}{{f}_{s}}}\right)$
$v={v}_{s}\left(\frac{{f}_{s}}{{f}_{s}-{f}_{-}}\right)$
$v=\left(1.5\frac{m}{s}\right)\left(\frac{0.30Hz}{0.30Hz-0.20Hz}\right)=4.5\frac{m}{s}$

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