At what speed in meters per second must a 145 g baseball be traveling to have a de Broglie wavelength of 0.500 nm?

Verified step by step guidance

1. First, convert the mass of the baseball from grams to kilograms for consistency in SI units. Recall that 1 g = 0.001 kg.
2. Use the de Broglie wavelength equation, \(\lambda = \frac{h}{mv}\), where \(\lambda\) is the wavelength, \(h\) is Planck's constant (\(6.626 \times 10^{-34} \text{ Js}\)), \(m\) is the mass of the object, and \(v\) is the velocity.
3. Substitute the given values into the de Broglie equation. Convert the wavelength from nanometers to meters by using the conversion factor (1 nm = \(1 \times 10^{-9}\) m).
4. Rearrange the equation to solve for \(v\), the velocity of the baseball: \(v = \frac{h}{m\lambda}\).
5. Calculate the velocity using the rearranged equation and the converted values.

Key Concepts

de Broglie Wavelength

The de Broglie wavelength is a concept in quantum mechanics that describes the wave-like behavior of particles. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant (6.626 x 10^-34 Js), and p is the momentum of the particle. This relationship implies that all matter exhibits wave properties, with the wavelength inversely proportional to momentum.

Momentum

Momentum is a physical quantity defined as the product of an object's mass and its velocity, expressed as p = mv. In the context of the de Broglie wavelength, momentum is crucial because it determines the wavelength of a particle. For a baseball, its mass must be converted to kilograms to maintain consistency in SI units when calculating momentum.

Unit Conversion

Unit conversion is the process of converting a quantity expressed in one set of units to another. In this problem, the mass of the baseball is given in grams, which must be converted to kilograms (1 g = 0.001 kg) to use in calculations involving SI units. Proper unit conversion is essential for accurate results in physics and chemistry calculations.