The wavelength of a golf ball can be determined using the de Broglie equation. By first finding the wavelength with known mass and velocity, and then using the desired wavelength to rearrange and solve for the unknown velocity, we apply the principles of wave-particle duality.
The wavelength of any object in motion can be calculated using the de Broglie equation λ = h/mv, where λ is the wavelength, h is Planck's constant (6.626 x 10-34 J·s), m is mass in kilograms, and v is velocity in meters per second.
For the golf ball with a mass of 46 grams (0.046 kg) moving at 30 m/s, the wavelength is calculated as λ = 6.626 x 10-34 J·s / (0.046 kg × 30 m/s).
To find the velocity when the wavelength is 5.6 x 10-3 nm, rearrange the equation to solve for v: v = h / (m·λ).
Using the de Broglie equation, we determine the wavelength for the initial conditions and then rearrange to find the velocity for the given wavelength.
Convert the mass of the golf ball to kilograms and the wavelength to meters when performing these calculations. After substituting values into the de Broglie equation, solve for the respective unknown.
Conclusion: The de Broglie equation allows us to understand the wave-particle duality by calculating the associated wavelength of any moving object, regardless of its size, demonstrating the fundamental principles of quantum mechanics even in macroscopic objects like a golf ball.