A particle is moving 5 times as fast as an electron.
Question:

A particle is moving 5 times as fast as an electron. The ratio of the de-Broglie wavelength of the particle to that of the electron is $1.878 \times 10^{-4}$. The mass of the particle is close to :

1. (1) $4.8 \times 10^{-27} \mathrm{~kg}$

2. (2) $9.1 \times 10^{-31} \mathrm{~kg}$

3. (3) $1.2 \times 10^{-28} \mathrm{~kg}$

4. (4) $9.7 \times 10^{-28} \mathrm{~kg}$

Correct Option: , 4

Solution:

(4)

de Broglie wavelength

$\lambda=\frac{h}{m v} \Rightarrow m=\frac{h}{\lambda v}$

Clearly, $m \propto \frac{1}{\lambda v}$

If $\lambda$ and $v$ be the wavelength and velocity of electron and $\lambda^{\prime}$ and $v^{\prime}$ be the wavelength and velocity of the particle then

$\Rightarrow \frac{m^{\prime}}{m}=\frac{v \lambda}{v^{\prime} \lambda^{\prime}}=\frac{1}{5} \times \frac{1}{1.878} \times 10^{-4}$

$\Rightarrow m=9.7 \times 10^{-28} \mathrm{~kg}$