Question

How fast must a \(56.5-g\) tennis ball travel to have a de Broglie wavelength equal to that of a photon of green light ( \(5400 \mathrm{~A}\) )?

Short answer

The tennis ball must travel at approximately \(2.16 \times 10^{-27} \mathrm{~m/s}\).

Step-by-step solution

Understand the de Broglie wavelength formula

The de Broglie wavelength is given by the formula \(\lambda = \frac{h}{p}\), where \(\lambda\) is the wavelength, \(h\) is Planck's constant \(6.626 \times 10^{-34} \mathrm{~J} \, \mathrm{s}\), and \(p\) is the momentum of the particle. Momentum \(p\) can be written as \(p = mv\), where \(m\) is the mass and \(v\) is the velocity.

Convert quantities to proper units

Convert the mass of the tennis ball from grams to kilograms: \(56.5 \mathrm{~g} = 0.0565 \mathrm{~kg}\). Also, convert the wavelength from \(\mathrm{~A}\) (Angstroms) to meters: \(5400 \mathrm{~A} = 5400 \times 10^{-10} \mathrm{~m} = 5.4 \times 10^{-7} \mathrm{~m}\).

Rearrange the de Broglie equation

Since we need to find the velocity \(v\), rearrange the de Broglie equation to solve for \(v\): \(\lambda = \frac{h}{mv} \Rightarrow v = \frac{h}{m\lambda}\).

Plug in the values

Insert the known values into the equation: \(v = \frac{6.626 \times 10^{-34} \mathrm{~J} \, \mathrm{s}}{(0.0565 \mathrm{~kg}) (5.4 \times 10^{-7} \mathrm{~m})}\).

Calculate the velocity

Perform the calculation: \(v = \frac{6.626 \times 10^{-34}}{0.0565 \times 5.4 \times 10^{-7}} \approx 2.16 \times 10^{-27} \mathrm{~m/s}\).

Interpret the result

The velocity obtained is extremely small, on the order of \(10^{-27} \mathrm{~m/s}\), indicating that the tennis ball would have to move incredibly slowly to have a de Broglie wavelength equal to that of a photon of green light.

Key concepts

Planck's constant

Planck's constant is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. It is a key element in the equation for the de Broglie wavelength. Planck's constant is denoted as \(h\) and is equal to \(6.626 \times 10^{-34} \text{ J} \times \text{s}\). This constant is crucial for calculating the wavelength of both particles and waves, bridging the gap between classical and quantum physics. By understanding Planck's constant, we can explore the wave-particle duality of matter, which is fundamental to quantum mechanics.

Momentum equation

Momentum is a measure of the motion of an object and is given by the product of its mass and velocity. For any object, the momentum \(p\) can be calculated using the equation \( p = mv \), where \(m\) is the object's mass and \(v\) is its velocity. In the context of the de Broglie wavelength, momentum helps bridge the gap between a particle's motion and its wave characteristics. By combining the momentum equation with the de Broglie formula \( \lambda = \frac{h}{p} \), we can derive the wavelength of a particle based on its mass and velocity.

Understanding this allows us to see that even objects with mass, like a tennis ball, exhibit wave-like properties under certain conditions.

Wavelength conversion

To solve problems involving wavelengths, it’s important to convert units to standard measurements—typically meters in scientific contexts. In the example, we converted the green light wavelength from angstroms to meters. Since \( 1 \text{ \AA} = 1 \times 10^{-10} \text{ m} \), the original 5400 \text{ \AA} becomes \( 5400 \times 10^{-10} \text{ m} = 5.4 \times 10^{-7} \text{ m} \). This conversion is essential for plugging into the de Broglie equation.

Similarly, converting the mass of the tennis ball from grams to kilograms ensures all quantities are in compatible units, simplifying calculations and reducing the chance of errors.

Mass and velocity relationship

The relationship between mass and velocity is crucial in understanding how we calculate the de Broglie wavelength. From the de Broglie equation \( \lambda = \frac{h}{mv} \), we see that the wavelength depends inversely on both mass and velocity. This means:

• Larger mass or higher velocity results in a shorter wavelength.
• Smaller mass or lower velocity yields a longer wavelength.

When solving for velocity, as in our tennis ball example, we rearrange the equation to \( v = \frac{h}{m\lambda} \). By inputting the known values of Planck's constant, mass, and wavelength, we can solve for the velocity, reflecting how these quantities are interdependent in determining the wave properties of matter.