Particle <math> <mi>A</mi> </math> <math></math> of mass <math> <msub> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mi>A</mi> </mrow> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </math> <math><msub><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></msub><mfrac><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></mfrac></math> moving along the <math> <mi>x</mi> </math> <math></math> -axis with velocity <math> <msub> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </math> <math><msub><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></msub></math> collides elastically with another particle <math></math> <math> <mi>B</mi> </math> at rest having mass <math> <msub> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mi>B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </mfrac> </math> <math><msub><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></msub><mfrac><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></mfrac></math> . If both particles move along the <math></math> <math> <mi>x</mi> </math> axis after the collision, the change <math> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </math> <math></math> in de-Broglie wavelength of particle <math> <mi>A</mi> </math> <math></math> , in terms of its de-Broglie wavelength <math><mfenced separators="|"><mrow><mrow><msub><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></msub></mrow></mrow></mfenced></math> <math> <mfenced separators="|"> <mrow> <mrow> <msub> <mrow> <mrow> <mi>λ</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfenced> </math> before collision is:

⇒ V ? 1 = 2 U c m ? - U 1 ? ⇒ 2 m / 2 V o m 2 + m 3 - V o ⇒ 6 5 V o - V o ⇒ V 0 5 ⇒ λ o = h c m 2 V o λ f = h c M 2 V o 5 ⇒ Δ λ = 8 h c m V o

Particle <math> <mi>A</mi> </math> <math></math>&nbsp;of mass <math> <msub> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mi>A</mi> </mrow> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </math> <math><msub><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></msub><mfrac><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></mfrac></math>&nbsp;moving along the <math> <mi>x</mi> </math> <math></math>&nbsp;-axis with velocity <math> <msub> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </math> <math><msub><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></msub></math>&nbsp;collides elastically with another particle&nbsp;<math></math> <math> <mi>B</mi> </math> at rest having mass <math> <msub> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mi>B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </mfrac> </math> <math><msub><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></msub><mfrac><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></mfrac></math>&nbsp;. If both particles move along the&nbsp;<math></math> <math> <mi>x</mi> </math> axis after the collision, the change <math> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </math> <math></math>&nbsp;in de-Broglie wavelength of particle <math> <mi>A</mi> </math> <math></math>&nbsp;, in terms of its de-Broglie wavelength&nbsp;<math><mfenced separators="|"><mrow><mrow><msub><mrow><mrow></mrow></mrow><mrow><mrow></mrow></mrow></msub></mrow></mrow></mfenced></math> <math> <mfenced separators="|"> <mrow> <mrow> <msub> <mrow> <mrow> <mi>λ</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfenced> </math> before collision is:

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Physics NCERT Exemplar Solutions Class 11th Chapter Ten Physics Class 12th Dual Nature of Radiation and Matter

Particle $A$ of mass $m_{A} = \frac{m}{2}$ $_{} \frac{}{}$ moving along the $x$ -axis with velocity $v_{0}$ $_{}$ collides elastically with another particle $B$ at rest having mass $m_{B} = \frac{m}{3}$ $_{} \frac{}{}$ . If both particles move along the $x$ axis after the collision, the change $Δ λ$ in de-Broglie wavelength of particle $A$ , in terms of its de-Broglie wavelength $(_{})$ $(λ_{0})$ before collision is:

Option 1 -

$Δ λ = \frac{3}{2} λ_{0}$

Option 2 -

$Δ λ = \frac{5}{2} λ_{0}$

Option 3 -

$Δ λ = 2 λ_{0}$

Option 4 -

$Δ λ = 4 λ_{0}$

5 Views | Posted 2 months ago

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Answered by

Vishal Baghel | Contributor-Level 10

2 months ago

Correct Option - 4

Detailed Solution:

$\Rightarrow {\overset{?}{V}}_{1} = 2 \overset{?}{U_{c m}} - \overset{?}{U_{1}} \Rightarrow 2 [\frac{m / 2 V_{o}}{\frac{m}{2} + \frac{m}{3}}] - V_{o}$

$\Rightarrow \frac{6}{5} V_{o} - V_{o} \Rightarrow \frac{V_{0}}{5}$

$\Rightarrow λ_{o} = \frac{h c}{(\frac{m}{2} V_{o})} λ_{f} = \frac{h c}{(\frac{M}{2} \frac{V_{o}}{5})}$

$\Rightarrow Δ λ = \frac{8 h c}{{m V}_{o}}$

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