Question

A particle $X$ moving with a certain velocity has a de-Broglie wavelength of $1A^0$. If the particle $Y$ has a mass of $25\%$ that of $X$ and velocity $75\%$ that of $X$, de Broglie's wavelength of $Y$ will be:

Answer 1

Step 1. Given data:

The de-Broglie wavelength of $X$ = $1A^0$.

Mass of $Y\left(m_Y\right)$ = $25\%$mass of $X$, ( $0.25m_X$)

Velocity of $Y$ $\left(v_Y\right)$ = $75\%$mass of $X$, ( $0.75v_X$)

Step 2. Formula used:

De Broglie wavelength $\lambda =\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$mv$}\right.$,

where $h=$plank's constant, $m,v$are mass and velocity respectively.
Step 3. Calculations:

${\lambda }_X=1A^0=\frac{h}{m_X.v_X}$

Also, ${\lambda }_Y=\frac{h}{m_Y\times v_Y}=\frac{h}{\left(0.25m_X\times 0.75v_X\right)}=5.33\frac{h}{m_X.v_X}$, ( By putting the values of $m_Y,v_Y$ )

Thus, ${\lambda }_Y=5.33A^0$, ( By putting the value of ${\lambda }_X$)

Hence, de Broglie's wavelength of $Y$ will be $5.33A^0$.