Question

A particle of mass 2 mg has the same wavelength as a neutron moving with a velocity of $ 3 \times 10^5 \, \text{ms}^{-1} $. The velocity of the particle is (mass of neutron is $ 1.67 \times 10^{-27} \, \text{Kg} $)

• A. \( 2.5 \times 10^{-16} \, \text{ms}^{-1} \)
• B. \( 1.5 \times 10^{-13} \, \text{ms}^{-1} \)
• C. \( 2.5 \times 10^{-13} \, \text{ms}^{-1} \)
• D. \( 1.5 \times 10^{-16} \, \text{ms}^{-1} \)

Hint:

When solving de Broglie wavelength problems, ensure to equate the wavelengths of both particles and solve for the unknown velocity.

Answer 1

The Correct Option is A

The de Broglie wavelength of a particle is given by the formula: \[ \lambda = \frac{h}{mv} \] Where: - \( h = 6.626 \times 10^{-34} \, \text{Js} \) is Planck's constant, - \( m \) is the mass of the particle, - \( v \) is the velocity of the particle. For the particle to have the same wavelength as the neutron, we have: \[ \frac{h}{mv} = \frac{h}{m_n v_n} \] Where: - \( m_n = 1.67 \times 10^{-27} \, \text{Kg} \) is the mass of the neutron, - \( v_n = 3 \times 10^5 \, \text{ms}^{-1} \) is the velocity of the neutron. Using the mass of the particle as \( m = 2 \times 10^{-3} \, \text{Kg} \) and equating the wavelengths: \[ \frac{6.626 \times 10^{-34}}{(2 \times 10^{-3})(v)} = \frac{6.626 \times 10^{-34}}{(1.67 \times 10^{-27})(3 \times 10^5)} \] Solving for \( v \): \[ v = \frac{1.67 \times 10^{-27} \times 3 \times 10^5}{2 \times 10^{-3}} = 2.5 \times 10^{-16} \, \text{ms}^{-1} \]
Thus, the velocity of the particle is \( 2.5 \times 10^{-16} \, \text{ms}^{-1} \).