Question

Calculate the wavelength, in nanometers, associated with a \(1.0 \times 10^{2}-\mathrm{g}\) golf ball moving at \(30 . \mathrm{m} / \mathrm{s}\) (about 67 mph). At what speed must the ball travel to have a wavelength of \(5.6 \times 10^{-3} \mathrm{nm} ?\)

Short answer

The golf ball's wavelength at 30 m/s is \(2.209 \times 10^{-25}\) nm. For a wavelength of \(5.6 \times 10^{-3}\) nm, the speed is \(1.183 \times 10^{-21}\) m/s.

Step-by-step solution

Understand the Formula & Constants

To find the wavelength associated with a moving object, we use the de Broglie wavelength formula:\[\lambda = \frac{h}{mv}\]where \( \lambda \) is the wavelength, \( h \) is Planck's constant \((6.626 \times 10^{-34} \text{ m}^2 \text{ kg} / \text{s})\), \( m \) is the mass of the object, and \( v \) is the velocity. In this problem, the mass \( m = 1.0 \times 10^{2} \text{ g} = 0.1 \text{ kg} \) and the velocity \( v = 30 \text{ m/s} \).

Calculate the Wavelength for Given Speed

Substitute the values into the de Broglie formula:\[\lambda = \frac{6.626 \times 10^{-34}}{0.1 \times 30}\]Calculate the expression by first finding the denominator: \( 0.1 \times 30 = 3 \). Then, divide Planck's constant by this product:\[\lambda = \frac{6.626 \times 10^{-34}}{3} = 2.209 \times 10^{-34} \text{ meters}\]Convert this result to nanometers by multiplying by \(10^9\), since 1 meter = \(10^9\) nm:\[\lambda = 2.209 \times 10^{-34} \times 10^9 = 2.209 \times 10^{-25} \text{ nm}\]

Rearrange Formula for Speed Calculation

Given that the golf ball needs to have a wavelength of \(5.6 \times 10^{-3} \text{ nm} \), first convert this to meters:\[5.6 \times 10^{-3} \text{ nm} = 5.6 \times 10^{-12} \text{ m}\]Rearrange the de Broglie equation to solve for velocity \( v \):\[v = \frac{h}{m\lambda}\]

Calculate the Speed for Given Wavelength

Substitute the values into the rearranged equation:\[v = \frac{6.626 \times 10^{-34}}{0.1 \times 5.6 \times 10^{-12}}\]Calculate the denominator first: \( 0.1 \times 5.6 \times 10^{-12} = 5.6 \times 10^{-13} \). Then solve:\[v = \frac{6.626 \times 10^{-34}}{5.6 \times 10^{-13}} = 1.183 \times 10^{-21} \text{ m/s}\]

Key concepts

Wavelength Calculation

The de Broglie wavelength is a crucial concept in understanding the wave-particle duality of matter. Louis de Broglie suggested that particles also have wave-like properties. To find the wavelength (\( \lambda \)) of an object, we use the formula \( \lambda = \frac{h}{mv} \). This equation requires knowing the mass \( m \), velocity \( v \), and Planck's constant \( h \).

• \( \lambda \) represents the wavelength of the particle.
• \( h \) is Planck's constant.
• \( m \) is the mass of the particle.
• \( v \) is the velocity at which the particle is moving.

For larger objects like a golf ball, their calculated de Broglie wavelength becomes incredibly small, often undetectable, yet this calculation demonstrates the theoretical applicability to all matter.

Planck's Constant

Planck's constant \( h \) is a fundamental quantity in quantum mechanics that helps express the scale at which quantum effects become significant.

• Its value is \( 6.626 \times 10^{-34} \text{ m}^2 \text{ kg} / \text{s} \).
• This constant relates the energy carried by a photon to its frequency.
• It is a cornerstone in the calculation of the de Broglie wavelength.

Planck's constant provides a bridge between the microworld of quantum particles and the macroworld we observe. It allows us to calculate wavelengths, as shown in the de Broglie equation, indicating the wave nature of particles.

Velocity

Velocity \( v \) refers to the speed of an object in a particular direction. It is a critical factor in the de Broglie wavelength calculation.

• In the formula \( \lambda = \frac{h}{mv} \), velocity directly affects the wavelength calculation.
• The higher the velocity of the particle, the shorter the de Broglie wavelength.

In scenarios involving quantum particles like electrons, even small changes in velocity can significantly impact the results. However, for larger objects like a golf ball, as in our exercise, very high velocities would be needed to make their de Broglie wavelength observable.

Mass Conversion

Mass conversion is essential when applying the de Broglie formula since it often requires the mass in kilograms. For this exercise:

• The golf ball's mass was given in grams, \( 1.0 \times 10^{2} \) g, which needs converting to kilograms before using the de Broglie equation.
• The conversion process is straightforward: since 1 kg equals 1000 g, convert by dividing by 1000.
• Thus, \( 1.0 \times 10^{2} \) g becomes 0.1 kg.

Accurate mass conversion is crucial in ensuring the correct calculation of wavelengths, particularly in quantum mechanics where precise measurements are key to valid results.