Question

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 \(35 \mathrm{~m} / \mathrm{s}(\sim 80 \mathrm{mi} / \mathrm{h})\)

Short answer

The de Broglie wavelength for an electron with a velocity 10% of the speed of light is approximately \(2.43\times 10^{-12}m\), and for a tennis ball served at 35m/s (80mi/h) with a mass of 55g (0.055kg), it is approximately \(3.42 \times 10^{-34}m\).

Step-by-step solution

Find the mass and velocity of each particle.

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: \[m = 55 g \times \frac{1 kg}{1000 g} = 0.055 kg\]

Calculate the de Broglie wavelength for each particle.

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 \(3.42 \times 10^{-34}m\).