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 41 m/s.

Steps to Solve

1. Understanding the de Broglie wavelength formula

The de Broglie wavelength ($\lambda$) is calculated using the formula: $$ \lambda = \frac{h}{p} $$ where $h$ is Planck's constant ($6.626 \times 10^{-34} , \text{Js}$) and $p$ is the momentum of the particle.

2. Calculate the momentum for the electron

The momentum ($p$) of the electron can be calculated as: $$ p = mv $$ where $m$ is the mass of the electron ($9.11 \times 10^{-31} , \text{kg}$) and $v$ is its velocity.

Since the velocity is 10% of the speed of light ($c = 3.00 \times 10^8 , \text{m/s}$): $$ v = 0.1 \times c = 0.1 \times 3.00 \times 10^8 , \text{m/s} = 3.00 \times 10^7 , \text{m/s} $$

Now, substituting into the momentum formula: $$ p = (9.11 \times 10^{-31} , \text{kg}) \times (3.00 \times 10^7 , \text{m/s}) $$

3. Calculate the de Broglie wavelength for the electron

Substituting $p$ into the de Broglie wavelength formula: $$ \lambda = \frac{h}{p} = \frac{6.626 \times 10^{-34} , \text{Js}}{p} $$

4. Calculate the momentum for the tennis ball

Convert the mass of the tennis ball from grams to kilograms: $$ m = 55 , \text{g} = 0.055 , \text{kg} $$ Now, calculate momentum using: $$ p = mv = 0.055 , \text{kg} \times 41 , \text{m/s} $$

5. Calculate the de Broglie wavelength for the tennis ball

Substituting $p$ into the de Broglie wavelength formula for the tennis ball: $$ \lambda = \frac{h}{p} = \frac{6.626 \times 10^{-34} , \text{Js}}{p} $$