Question

What is the de Broglie wavelength of a \(2.000 \cdot 10^{3}-\mathrm{kg}\) car moving at a speed of \(100.0 \mathrm{~km} / \mathrm{h} ?\)

Short answer

Answer: The de Broglie wavelength of the car is approximately 1.19 × 10⁻³⁸ meters.

Step-by-step solution

Convert speed from km/h to m/s

To convert the speed of the car from 100.0 km/h to m/s, we need to multiply it by a conversion factor. There are 1000 meters in a kilometer and 3600 seconds in an hour, so the conversion factor is: \(\text{conversion factor} = \dfrac{1000 \text{ meters}}{1 \text{ kilometer}} \times \dfrac{1 \text{ hour}}{3600 \text{ seconds}}\) Now, we can convert the speed: \(v = 100.0 \dfrac{\text{km}}{\text{h}} \times \text{conversion factor} = 100.0 \dfrac{\text{km}}{\text{h}} \times \dfrac{1000 \text{ meters}}{1 \text{ kilometer}} \times \dfrac{1 \text{ hour}}{3600 \text{ seconds}} = 27.8\ \dfrac{\text{m}}{\text{s}}\)

Calculate momentum

Now that we have the speed of the car in m/s, we can calculate its momentum using the formula: \(p = mv\) where \(m = 2.000 \times 10^{3}\ \text{kg}\) (mass of the car) and \(v = 27.8\ \dfrac{\text{m}}{\text{s}}\) (speed of the car): \(p = (2.000 \times 10^{3}\ \text{kg})(27.8\ \dfrac{\text{m}}{\text{s}}) = 5.56 \times 10^{4}\ \text{kg}\cdot \text{m} / \text{s}\)

Compute the de Broglie wavelength

Finally, we can use the formula, \(\lambda = \dfrac{h}{p}\), to calculate the de Broglie wavelength. Planck's constant, \(h\), is approximately \(6.63 \times 10^{-34}\ \text{J}\cdot\text{s}\). We can rewrite Planck's constant in units of \(\text{kg}\cdot\text{m}^2/\text{s}\) (as \(1\ \text{J} = 1\ \text{kg}\cdot\text{m}^2/\text{s}^2\)): \(h = 6.63 \times 10^{-34}\ \text{kg}\cdot\text{m}^2/\text{s}\) Now, we can find the de Broglie wavelength: \(\lambda = \dfrac{h}{p} = \dfrac{6.63 \times 10^{-34}\ \text{kg}\cdot\text{m}^2/\text{s}}{5.56 \times 10^{4}\ \text{kg}\cdot\text{m}/\text{s}} = 1.19 \times 10^{-38}\ \text{m}\) The de Broglie wavelength of the car is approximately \(1.19 \times 10^{-38}\) meters.

Key concepts

Momentum Calculation

Understanding momentum is key to calculating the de Broglie wavelength of an object.
Momentum, denoted as \( p \), is defined as the product of an object's mass \( m \) and its velocity \( v \). The formula for momentum is expressed as \( p = mv \).
This relationship helps us understand how much motion an object has and is crucial when delving into quantum mechanics.

For instance, in the exercise where we need to calculate the de Broglie wavelength of a car, the mass of the car is given as \( 2.000 \times 10^3 \) kg. After converting the speed from km/h to m/s, we use this speed to find the momentum.
By multiplying the mass \( 2.000 \times 10^3 \) kg by the speed \( 27.8 \dfrac{\text{m}}{\text{s}} \), we calculate the momentum \( 5.56 \times 10^4 \ \text{kg} \cdot \dfrac{\text{m}}{\text{s}} \).
This calculated momentum is essential for determining the wavelength using de Broglie's formula.

Planck's Constant

Planck's constant is a fundamental constant in physics, represented by the symbol \( h \). It has a pivotal role in quantum mechanics and helps in the study of particle-wave duality.
Planck's constant has a value of approximately \( 6.63 \times 10^{-34} \ \text{J} \cdot \text{s} \).
In our exercise, it gets converted into the units \( \text{kg} \cdot \text{m}^2/\text{s} \), which are more suitable for our calculations as \( 1 \ \text{J} = 1 \ \text{kg} \cdot \text{m}^2/\text{s}^2 \).
This constant is essential for connecting the particle's momentum to its de Broglie wavelength using the equation \( \lambda = \frac{h}{p} \).

By dividing Planck's constant by momentum, we determine the wavelength of an object in motion.

• Planck's constant provides a bridge between the macroscopic world and the quantum realm.
• It allows even massive objects like cars to be analyzed using quantum mechanics.
• Despite its small value, it holds significant implications for our understanding of matter and energy interactions.

Speed Conversion

Speed conversion is a fundamental skill often required in physics to ensure that all quantities fit the necessary units for calculations.
In our exercise, we began with the car's speed given in kilometers per hour (km/h), but to use physics formulas, the standard unit of meters per second (m/s) is required.

To convert from km/h to m/s:

• Multiply by \( \frac{1000}{1} \), which converts kilometers to meters.
• Multiply by \( \frac{1}{3600} \), which converts hours to seconds.

This conversion results in a speed of \( 27.8 \dfrac{\text{m}}{\text{s}} \).
Accurate conversion is essential because it ensures the compatibility of units across calculations. Incorrect speed units can lead to erroneous results.

• Conversion ensures uniformity and accuracy in calculations.
• This step is crucial in problems involving various physical quantities.
• Proper conversion also aids in global standardization of scientific measurements.