"The de Broglie wavelengths of a proton and an $\alpha$ particle are $\lambda$ and $2 \lambda$ respectively. The ratio of the velocities of proton and $\alpha$ particle will be :"

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To find the ratio of velocities of two particles based on their de Broglie wavelengths, we can use the de Broglie wavelength formula, which relates the momentum of a particle to its wavelength. The de Broglie's wavelength formula is given by:

$\lambda = \frac{h}{p}$

where:

$\lambda$ is the de Broglie wavelength,

$h$ is the Planck constant, and

$p$ is the momentum of the particle.

The momentum $p$ of a particle can also be expressed as the product of its mass $m$ and velocity $v$:

$p = mv$

So, we can rewrite the de Broglie wavelength equation in terms of mass and velocity:

$\lambda = \frac{h}{mv}$

For the proton (let's use subscript $p$ for proton), the wavelength is $\lambda$:

$\lambda_p = \frac{h}{m_p v_p}$

For the $\alpha$ particle (let's use subscript $\alpha$ for alpha), the wavelength is $2\lambda$:

$2\lambda = \frac{h}{m_\alpha v_\alpha}$

We are interested in finding the ratio of the velocities $\frac{v_p}{v_\alpha}$. Using the given data about wavelengths:

$\lambda_p = \lambda$

$\lambda_\alpha = 2\lambda$

Using the de Broglie equation for both particles:

$\frac{h}{m_p v_p} = \lambda$

$\frac{h}{m_\alpha v_\alpha} = 2\lambda$

Dividing the second equation by the first equation gives us:

$\frac{\frac{h}{m_\alpha v_\alpha}}{\frac{h}{m_p v_p}} = \frac{2\lambda}{\lambda}$

$\frac{m_p v_p}{m_\alpha v_\alpha} = \frac{2}{1}$

$\frac{v_p}{v_\alpha} = \frac{2m_\alpha}{m_p}$

Since we know an $\alpha$ particle consists of 2 protons and 2 neutrons (essentially four nucleons), the mass of an $\alpha$ particle is roughly four times the mass of a proton ($m_\alpha \approx 4m_p$).

Substituting $m_\alpha$ with $4m_p$ in the equation:

$\frac{v_p}{v_\alpha} = \frac{2(4m_p)}{m_p}$

$\frac{v_p}{v_\alpha} = 2 \cdot 4$

$\frac{v_p}{v_\alpha} = 8$

Therefore, the ratio of the velocities of proton to $\alpha$ particle is $8:1$, which corresponds to Option A.

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

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

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