A smooth sphere is moving on a horizontal surface with velocity vector&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mover accent="true"><mi mathvariant="normal">i</mi><mo>¯</mo></mover><mo>+</mo><mn>2</mn><mover accent="true"><mi mathvariant="normal">j</mi><mo>¯</mo></mover></math>&nbsp;immediately before it hits a vertical wall. The wall is parallel to&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mi mathvariant="normal">j</mi><mo>¯</mo></mover></math>&nbsp;vector and the coefficient of restitution between the sphere and the wall is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">e</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>. The velocity vector of the sphere after it hits the wall is

A particle of mass 3m at rest decays into two particles of masses m&nbsp;and 2m&nbsp;having non-zero velocities. The ratio of the de Broglie wavelengths of the particles (<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>λ</mi><mn>1</mn></msub><msub><mi>λ</mi><mn>2</mn></msub></mfrac></math>) is

An electron is accelerated through a potential difference of 1600 V. Its de Broglie wavelength is, (nearly)&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle mathsize="16px"><mfenced><mrow><msub><mi mathvariant="normal">m</mi><mi mathvariant="normal">e</mi></msub><mo>=</mo><mn>9</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>31</mn></mrow></msup><mo>&nbsp;</mo><mi>kg</mi></mrow></mfenced></mstyle></math>

&nbsp;De Broglie’s relation is:

A particle moving in the positive X direction has de Broglie wavelength&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub></math>. Another particle also moving in the same direction has de Broglie wavelength&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>2</mn></msub></math>. The two particles make a collision and after the collision they move together as a single particle. The de Broglie wavelength of this single particle is given by&nbsp;

An electron (mass m) with initial velocity&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mi mathvariant="normal">ν</mi><mo>→</mo></mover><mo>=</mo><msub><mi mathvariant="normal">v</mi><mn>0</mn></msub><mover accent="true"><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><msub><mi mathvariant="normal">v</mi><mn>0</mn></msub><mover accent="true"><mi mathvariant="normal">j</mi><mo>^</mo></mover></math>&nbsp; is in an electric field &nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mi>E</mi><mo>→</mo></mover><mo>=</mo><mo>−</mo><msub><mi>E</mi><mn>0</mn></msub><mover accent="true"><mi>k</mi><mo>^</mo></mover><mo>.</mo></math>&nbsp;If &nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>0</mn></msub></math>&nbsp;is initial de-Broglie wavelength of electron, its de-Broglie wavelength at time t is given by :

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A smooth sphere is moving on a horizontal surface with velocity vector 2i¯+2j¯ immediately before it hits a vertical wall. The wall is parallel to j¯ vector and the coefficient of restitution between the sphere and the wall is e=12. The velocity vector of the sphere after it hits the wall is

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