Question

Just as light waves have particle behavior, a moving particle has a wave nature. The faster the particle is moving, the higher its kinetic energy and the shorter its wavelength. The wavelength, λ, of a particle of mass m, and moving at velocity v, is given by the de Broglie relation

λ=hmv

where h=6.626×10−34 J⋅s is Planck's constant.

This formula applies to all objects, regardless of size, but the de Broglie wavelength of macro objects is miniscule compared to their size, so we cannot observe their wave properties. In contrast, the wave properties of subatomic particles can be seen in such experiments as diffraction of electrons by a metal crystal.

Part A

The mass of an electron is 9.11×10−31 kg. If the de Broglie wavelength for an electron in a hydrogen atom is 3.31×10−10 m, how fast is the electron moving relative to the speed of light? The speed of light is 3.00×108 m/s.

Express your answer numerically as a percent.

Part B

The mass of a golf ball is 45.9 g . If it leaves the tee with a speed of 60.0 m/s , what is its corresponding wavelength?

Answer 1

A)

The Equation is given by

λ= h/(mv)

where

λ= Wavelength

h = Planck Constant = 6.626*10^-34 J s

m = mass in kg

v = velocity in m/s

Now, substitute all data

λ= h/(mv)

v = h/(m*WL) = (6.626*10^-34 ) / ((9.11*10^-31)(3.31*10^-10))

v = 2197379.46084 m/s

v light = 3*10^8 m/s

relative % = --> V / Vlight * 100% = 2197379.46084 / (3*10^8) * 100 = 0.732 %

B)

mass = 45.9 g = 45.9*10^-3 kg

v = 60 m/s

λ= h/(mv)

where

λ= Wavelength

h = Planck Constant = 6.626*10^-34 J s

m = mass in kg

v = velocity in m/s

Now, substitute all data

λ= h/(mv)

WL = (6.626*10^-34 ) /((45.9*10^-3)(60))

WL = 2.405*10^-34 m