## Q. 7.PS.5

Tennis Balls and Electrons

At Wimbledon, tennis serves routinely reach more than 100 mi/h. Compare the de Broglie wavelength (nm) of an electron moving at a velocity of $5.0 × 10^6 m/s$ with that of a tennis ball traveling at $56.0 m/s (125 mi/h)$. Masses: electron = $9.11 × 10^{-31} kg$; tennis ball = $0.0567 kg$.

## Verified Solution

The wavelength of the electron is much longer than that of the tennis ball: electron = $0.15 nm$; tennis ball = $2.09 × 10^{-25} nm$.

Strategy and Explanation We can substitute the mass and velocity into the de Broglie wave equation to calculate the corresponding wavelength. Planck’s constant, $b$, is

$6.626 × 10^{-34} J⋅s$, and $1 J = \frac{1 kg⋅m^2}{s^2}$ so that $b = 6.626 × 10^{-34} kg⋅m^2 s^{-1}.$

For the electron:

$λ = \frac{6.626 × 10^{-34} kg⋅m^2 s^{-1}}{(9.11 × 10^{-31} kg)(5.0 × 10^6 m/s)} = 1.5 × 10^{-10} m × \frac{1 nm}{10^{-9} m} = 0.15 nm$

For the tennis ball:

$λ = \frac{6.626 × 10^{-34} kg⋅m^2 s^{-1}}{(0.0567 kg)(56.0 m/s)} = 2.09 × 10^{-34} m × \frac{1 nm}{10^{-9} m} = 2.09 × 10^{-25} nm$

The wavelength of the electron is in the X-ray region of the electromagnetic spectrum (Figure 7.1, ← p. 222). The wavelength of the tennis ball is far too short to observe.