Question

The de Broglie wavelengths of two fast-moving particles \( X \), \( Y \) are \( 1 \) nm, \( 3 \) nm respectively. Mass of \( X \) is nine times the mass of \( Y \). The ratio of kinetic energies of \( X \), \( Y \) is:

• A. \( 1 : 3 \)
• B. \( 1 : 1 \)
• C. \( 9 : 1 \)
• D. \( 1 : 9 \)

Hint:

For kinetic energy based on de Broglie wavelength: \[ E \propto \frac{1}{m\lambda^2} \] Adjust proportions accordingly when comparing different particles.

Answer 1

The Correct Option is C

Step 1: de Broglie Wavelength Formula The de Broglie wavelength is given by: \[ \lambda = \frac{h}{\sqrt{2mE}} \] Rearranging for kinetic energy \( E \): \[ E = \frac{h^2}{2m\lambda^2} \] Step 2: Computing Ratio \[ \frac{E_X}{E_Y} = \frac{m_Y \lambda_Y^2}{m_X \lambda_X^2} \] Given: - \( m_X = 9 m_Y \), - \( \lambda_X = 1 \) nm, \( \lambda_Y = 3 \) nm. Substituting: \[ \frac{E_X}{E_Y} = \frac{m_Y (3)^2}{9m_Y (1)^2} \] \[ = \frac{9}{9} = 1 \] Conclusion Thus, the correct answer is: \[ 9 : 1 \]