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Chapter 7: Problem 59

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 g) served at 35 $\mathrm{m} / \mathrm{s}(\sim 80 \mathrm{mi} / \mathrm{h})$

Short Answer

Expert verified
The de Broglie wavelength for an electron moving at 10% the speed of light is determined by using the formula \(\lambda = \frac{h}{p}\), where \(h = 6.626 \times 10^{-34} \, \text{Js}\) and \(p = (9.11 \times 10^{-31} \, \text{kg})(0.1 \times 3 \times 10^8 \, \text{m/s})\). Calculating the de Broglie wavelength yields \(\lambda_e \approx 7.29 \times 10^{-12} \, \text{m}\). For the tennis ball served at 35 m/s, we again use the formula \(\lambda = \frac{h}{p}\), where \(p = (0.055 \, \text{kg}) \times (35 \, \text{m/s})\). Calculating the de Broglie wavelength results in \(\lambda_t \approx 3.46 \times 10^{-34} \, \text{m}\).

Step by step solution

01

Case a: Electron moving at 10% of the speed of light

First, we need to find the momentum of the electron. The mass of an electron is \(9.11 \times 10^{-31} \, \text{kg}\), and its velocity is 10% of the speed of light, \(c = 3 \times 10^8 \, \text{m/s}\). Therefore, the velocity of the electron is: \[v = 0.1c = 0.1 \times 3 \times 10^8 \, \text{m/s}\] The momentum, \(p\), of the electron can be calculated as: \[p = mv = (9.11 \times 10^{-31} \, \text{kg}) \times (0.1 \times 3 \times 10^8 \, \text{m/s})\] Now, we can calculate the de Broglie wavelength as: \[\lambda = \frac{h}{p}\]
02

Case b: Tennis ball served at 35 m/s

In this case, we have a tennis ball with a mass of 55 g which is equal to \(0.055\) kg. The velocity of the tennis ball is given as 35 m/s. First, we find the momentum of the tennis ball: \[p = mv = (0.055 \, \text{kg}) \times (35 \, \text{m/s})\] Then, we can calculate the de Broglie wavelength as: \[\lambda = \frac{h}{p}\] By calculating the de Broglie wavelength for both cases, we will have the answers to the given exercise.

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