Calculate the de Broglie wavelength for each of the following. a. an electron with a velocity \(10 . \%\) of the speed of light b. a tennis ball \((55 \mathrm{~g})\) served at \(35 \mathrm{~m} / \mathrm{s}(\sim 80 \mathrm{mi} / \mathrm{h})\)

Firstly, we will find the mass and velocity of each particle. a. For the electron, we are given its velocity as 10% of the speed of light. The mass of the electron is \(9.11 \times 10^{-31} kg\), and the speed of light is \(3 \times 10^8 m/s\). Thus, the electron's velocity is: \[v = 0.1 \times (3 \times 10^8 m/s) = 3 \times 10^7 m/s\] b. For the tennis ball, we have its mass as \(55g\) and its velocity as \(35m/s\). To calculate the mass in kilograms, we convert as follows: \[m = 55 g \times \frac{1 kg}{1000 g} = 0.055 kg\]

Now we will use the de Broglie wavelength formula for each particle using their respective masses and velocities. a. For the electron with mass \(m = 9.11 \times 10^{-31} kg\) and velocity \(v = 3 \times 10^7 m/s\), we will use Planck's constant \(h = 6.626 \times 10^{-34} Js\): \[\lambda_e = \frac{h}{mv} = \frac{6.626 \times 10^{-34} Js}{(9.11 \times 10^{-31} kg)(3 \times 10^7 m/s)}\approx 2.43\times 10^{-12}m\] So, the de Broglie wavelength of the electron is approximately \(2.43\times 10^{-12}m\). b. For the tennis ball with mass \(m = 0.055 kg\) and velocity \(v = 35 m/s\), we will also use Planck's constant \(h = 6.626 \times 10^{-34} Js\): \[\lambda_t = \frac{h}{mv} = \frac{6.626 \times 10^{-34} Js}{(0.055 kg)(35 m/s)}\approx 3.42 \times 10^{-34}m\] So, the de Broglie wavelength of the tennis ball is approximately \(3.42 \times 10^{-34}m\).