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:
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:
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For kinetic energy based on de Broglie wavelength:
\[
E \propto \frac{1}{m\lambda^2}
\]
Adjust proportions accordingly when comparing different particles.
Updated On: Mar 18, 2026
- \( 1 : 3 \)
- \( 1 : 1 \)
- \( 9 : 1 \)
- \( 1 : 9 \)
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The Correct Option is C
Solution and Explanation
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:
\[
9 : 1
\]
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