Question

An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy 10 keV, and the second with 100 keV. Which is faster, the electron or the proton ? Obtain the ratio of their speeds. (electron mass = 9.11×10¯³¹ kg, proton mass = 1.67×10¯²⁷ kg, 1 eV = 1.60 ×10¯¹⁹ J).

Answer 1

Given:

The kinetic energy of the electron, Ke =10 keV, the kinetic energy of the proton, K p =100 keV, the mass of the electron, m e =9.11× 10 −31  kg and the mass of the proton, m p =1.67× 10 −27  kg.

The kinetic energy of the electron can be given as, $K_e=\frac{1}{2}{m}_e{v}_e^2,v_e=\sqrt{\frac{2K_e}{m_e}}$ ...(1)

The kinetic energy of the proton can be given as, $K_p=\frac{1}{2}{m}_p{v}_p^2,v_p=\sqrt{\frac{2K_p}{m_p}}$ ...(2)

By dividing equation (1) by equation (2), we get

$\frac{v_e}{v_p}=\frac{\sqrt{2K_e{m}_p}}{\sqrt{2K_p{m}_e}}$

By substituting the given values in the above equation, we get

$\frac{v_e}{v_p}=\frac{\sqrt{2\left(10\right)\times 1.67\times {10}^{-27}}}{\sqrt{2\left(100\right)\times 9.11\times {10}^{-31}}}$

$v_e=13.5v_p$

The above relation shows that the electron is moving faster than the proton.