The de Broglie wavelengths of a proton and an α particle are \( \lambda \) and \( 2\lambda \) respectively. The ratio of the velocities of proton and α particle will be:

Question

The de-Broglie wavelength of a moving bus with speed \( v \) is \( \lambda \). Some passengers left the bus at a stoppage. Now when the bus moves with twice of its initial speed, its kinetic energy is found to be twice its initial value. What is the de-Broglie wavelength of the bus now?

Answer 1

The de Broglie wavelength equation is defined as:

\[ \lambda = \frac{h}{p} = \frac{h}{mv} \]

with the following parameters:

\(h\) represents Planck’s constant.
\(m\) denotes the mass of the particle.
\(v\) signifies the velocity.

For a proton and an \(\alpha\)-particle, the wavelength ratio is:

\[ \frac{\lambda_p}{\lambda_\alpha} = \frac{m_\alpha v_\alpha}{m_p v_p} \]

Given that \(m_\alpha = 4m_p\) (an \(\alpha\)-particle has four times the mass of a proton) and the specified velocity-wavelength relationship, the ratio of their velocities is determined to be:

\[ v_p : v_\alpha = 8 : 1 \]

Therefore, Option (4) is the correct choice.

The de Broglie wavelengths of a proton and an α particle are \( \lambda \) and \( 2\lambda \) respectively. The ratio of the velocities of proton and α particle will be:

The Correct Option is D

Solution and Explanation

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