Question

The wavelength associated with a golf ball weighing $200{\text{g}}$ and moving at a speed of $5{\text{m}}{{\text{h}}^{ - 1}}$ is of the order:
A.${10^{ - 10}}{\text{m}}$
B.${10^{ - 20}}{\text{m}}$
C.${10^{ - 30}}{\text{m}}$
D.${10^{ - 40}}{\text{m}}$

Answer 1

de-Broglie wavelength can be calculated from its momentum, mass and kinetic energy of the particle of matter. Also, de-Broglie wavelength is determined for matter which explains wave nature of matter like electrons, protons etc.
Formula Used:
$\lambda = \dfrac{{\text{h}}}{{\text{p}}} = \dfrac{{\text{h}}}{{\sqrt {2{\text{mKE}}} }}$
Here h represents Planck’s constant, p represents momentum, m represents mass, v is velocity, KE is kinetic energy of particles of matter.

Complete step by step answer:
As light has dual nature. Light possesses both particle and wave nature. De-Broglie suggested that matter could also have both nature; particles and wave nature. As we all already know, particle nature and properties of particles of matter like they have space between them, particles of a matter attract each other; they show movement about their mean position. But de-Broglie suggested the matter behavior of matter and explained it as matter waves. So, matter waves are the imaginary waves associated with material or matter particles. The wavelength of matter waves is known as de-Broglie wavelength.
Mathematical representation of de-Broglie wavelength in terms of momentum, kinetic energy can be given as:
$\lambda = \dfrac{{\text{h}}}{{\text{p}}} = \dfrac{{\text{h}}}{{{\text{mv}}}}$ and
$\lambda = \dfrac{{\text{h}}}{{\text{p}}} = \dfrac{{\text{h}}}{{\sqrt {2{\text{mKE}}} }}$ ;
Here h represents Planck’s constant, p represents momentum, m represents mass, v is velocity, KE is kinetic energy of particles of matter.
As given in question, mass of matter or golf ball is $200{\text{g}}$ and its speed is given as $5{\text{m}}{{\text{h}}^{ - 1}}$ . As we already know Planck’s constant and has the value $6.6 \times {10^{ - 34}}{\text{Jsec}}$ , but we have to convert given mass in kilogram and given speed in ${\text{m}}{{\text{s}}^{ - 1}}$ .
Therefore, we get mass of ball $0.2{\text{kg}}$ and its speed $\dfrac{5}{{3600}}{\text{m}}{{\text{s}}^{ - 1}}$ .
Thus, wavelength can be given as:
$\lambda = \dfrac{{\text{h}}}{{{\text{mv}}}} = \dfrac{{6.6 \times {{10}^{ - 34}}}}{{0.2 \times \left( {\dfrac{5}{{3600}}} \right)}} = 2.4 \times {10^{ - 30}}{\text{m}}$ .

Thus, the correct option is C.

Note:
de-Broglie wavelength helps to determine the wave associated with the matter, which helps to conclude the dual nature of matter like light also has dual nature. Planck theory explains the particle nature of light. He explained the concept of quantum or photon and energy of quantum to be directly proportional to its frequency associated.