Question

calculate the wavelengths of the following objects.
2nd attempt
part 1 (1 point)
a 75 kg athlete running a 5.0 - minute mile:
×10 m
part 2 (1 point)
earth (mass = 6.0×10²⁷g) moving through space at 2.7×10⁴ m/s:
×10 m
1st attempt
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Explanation:

Step1: Calculate the velocity of the athlete

First, convert the time for a mile (1 mile = 1609.34 m) from minutes to seconds. \(t = 5.0\ \text{min}\times60\ \text{s/min}=300\ \text{s}\).
The velocity \(v=\frac{d}{t}=\frac{1609.34\ \text{m}}{300\ \text{s}}\approx5.36\ \text{m/s}\).

Step2: Use the de - Broglie wavelength formula \(\lambda=\frac{h}{mv}\)

where \(h = 6.626\times10^{-34}\ \text{J}\cdot\text{s}\), \(m = 75\ \text{kg}\), and \(v\approx5.36\ \text{m/s}\).
\(\lambda=\frac{6.626\times10^{-34}}{75\times5.36}\)
\(=\frac{6.626\times10^{-34}}{402}\approx1.65\times10^{-36}\ \text{m}\)

Step3: Convert the mass of the Earth to kg

\(m = 6.0\times10^{27}\ \text{g}=6.0\times10^{24}\ \text{kg}\)

Step4: Use the de - Broglie wavelength formula \(\lambda=\frac{h}{mv}\) for the Earth

where \(h = 6.626\times10^{-34}\ \text{J}\cdot\text{s}\), \(m = 6.0\times10^{24}\ \text{kg}\), and \(v = 2.7\times10^{4}\ \text{m/s}\)
\(\lambda=\frac{6.626\times10^{-34}}{6.0\times10^{24}\times2.7\times10^{4}}\)
\(=\frac{6.626\times10^{-34}}{1.62\times10^{29}}\approx4.09\times10^{-63}\ \text{m}\)

Answer:

Part 1: \(1.65\times 10^{-36}\ \text{m}\)
Part 2: \(4.09\times 10^{-63}\ \text{m}\)