Question

A proton has a kinetic energy of 7.5 MeV. What is the speed of an alpha particle that has the same DeBroglie wavelength as the proton? The mass of the alpha particle is 6.65E-27 kg. O 9517 m/s 9.53E6 m/s 1833 m/s O 2.38E13 m/s O 0.747 m/s

Answer 1

Answer : ◉ 9.53E6 m/s

Solution :

Given :

Kinetic energy of proton : KE = 7.5 MeV = 7.5 x 10⁶ eV = (7.5 x 10⁶ eV)(1.6 x 10^-19 J/eV) = 1.2 x 10^-12 J

Mass of proton is : m_p = 1.67 x 10^-27 kg

and, Mass of alpha particle is : m_a = 6.65 x 10^-27 kg

.

Here De-Broglie wavelength of proton will be given by :

$\lambda _p=\frac{h}{m_pv}=\frac{h}{\sqrt{2m_p\cdot KE}}$

And, De-Broglie wavelength of alpha particle will be :

$\lambda _a=\frac{h}{m_av}$

Since, DeBroglie wavelength of alpha particle is equal to the Debroglie wavelength of Proton.

$\therefore \ \frac{h}{\sqrt{2m_p\cdot KE}}=\frac{h}{m_av}$

$\therefore \ \frac{1}{\sqrt{2m_p\cdot KE}}=\frac{1}{m_av}$

$\therefore \ m_av=\sqrt{2m_p\cdot KE}$

$\therefore \ v=\frac{\sqrt{2m_p\cdot KE}}{m_a}$

$\therefore \ v=\frac{\sqrt{2(1.67\times 10^{-27}\ kg)\cdot (1.2\times 10^{-12}\ J)}}{(6.65\times 10^{-27}\ kg)}$

$\mathbf{\therefore \ v=9.53\times 10^6\ m/s}$