Question

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

Answer 1

$\text{Given}:$

$m_e=9.1\times {10}^{-31}kg,$

$m_p=1.6\times {10}^{-27}kg,$

$K_e=10KeV,K_p=100KeV$

Here, kinetic energies of electron and proton is given so take the ratio of their respective energies to arrive at the ratio of their velocities.

$\frac{K_e}{K_m}=\frac{\frac{1}{2}{m}_e{v}_e^2}{\frac{1}{2}{m}_p{v}_p^2}$

$=\frac{\frac{1}{2}\times 9.1\times {10}^{-31}\times v_e^2}{\frac{1}{2}\times 1.6\times {10}^{-27}\times v_p^2}$

$=\frac{9.1\times {10}^{-31}\times v_e^2}{1.6\times {10}^{-27}\times v_p^2}=\frac{1}{10}$

Therefore,

$\frac{v_e^2}{v_p^2}=0.176\times {10}^3$

So, that gives,

$\frac{v_e}{v_p}=13.26$

The mass of the electron is much lesser than that of proton, so it is faster than proton.