Question

The velocity associated with a proton moving in a potential difference of 1000 V is $4.37\times {{10}^{5}}m{{s}^{-1}}$. If the hockey ball of mass of 0.1 kg is moving with this velocity, calculate the wavelength associated with this velocity.

Answer 1

Hint: As we know, the de Broglie wavelength is an important concept while learning quantum mechanics. The wavelength which is associated with an entity in relation to its mass and momentum is known as the de Broglie wavelength. We could find the wavelength by dividing the Planck's constant by the product of mass and velocity of the particle.

Complete step by step answer:
  - As we know, matter has a dual nature of wave-particles. The de Broglie waves was named after Louis de Broglie .It is the property of an object that varies in space or time while behaving similar to waves and it’s also called matter-waves.
  - In the question we are asked to find the wavelength of a hockey ball of mass of 0.1 kg which is moving with the velocity$4.37\times {{10}^{5}}m{{s}^{-1}}$. The wavelength can be found by using De-Broglie wavelength.

- The De-Broglie wavelength can be calculated by using the following formula:
\[\lambda =\dfrac{h}{mv}\]
Where $\lambda $is the wavelength of object
             h is the Planck’s constant ( $6.626\times {{10}^{-34}}Js$)
             m is the mass ( 0.1 kg)
             $v$ is the velocity (given as $4.37\times {{10}^{5}}m{{s}^{-1}}$)

Substitute this values in the above equation, we get
\[\lambda =\dfrac{6.626\times {{10}^{-34}}Js}{0.1Kg\times 4.37\times {{10}^{5}}m{{s}^{-1}}}=1.516\times {{10}^{-28}}m\]

Therefore the wavelength of a hockey ball of mass of 0.1 kg which is moving with the velocity$4.37\times {{10}^{5}}m{{s}^{-1}}$ is $1.516\times {{10}^{-28}}m$.

Note: Since momentum is obtained by multiplying mass by velocity, the de Broglie wavelength is inversely proportional to momentum. In order to find the de Broglie wavelength in terms of kinetic energy we could use the equation $\lambda =\dfrac{h}{\sqrt{2m{{E}_{k}}}}$ where ${{E}_{k}}$ is the kinetic energy of particle which we need to find the wavelength.