### Q.12:- 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:-**Electron is faster; Ratio of speeds is 13.54 : 1

Mass of the electron, *m*_{e} = 9.11 × 10^{–31} kg

Mass of the proton, *m*_{p} = 1.67 × 10^{– 27} kg

Kinetic energy of the electron, *E*_{Ke} = 10 keV = 10^{4} eV

= 10^{4} × 1.60 × 10^{–19}

= 1.60 × 10^{–15} J

Kinetic energy of the proton, *E*_{Kp} = 100 keV = 10^{5} eV = 1.60 × 10^{–14} J

For the velocity of an electron v_{e}, its kinetic energy is given by the relation:

*E*_{Ke} = (1/2) mv_{e}^{2}

∴ v_{e} = (2E_{Ke} / m)^{1/2}

= (2 × 1.60 × 10^{-15} / 9.11 × 10^{-31})^{1/2} = 5.93 × 10^{7} m/s

For the velocity of a proton v_{p}, its kinetic energy is given by the relation:

*E*_{Kp} = (1/2)mv_{p}^{2}

*v*_{p} = (2 × 1.6 × 10^{-14} / 1.67 × 10^{-27} )^{1/2} = 4.38 × 10^{6} m/s

Hence, the electron is moving faster than the proton.

The ratio of their speeds

*v*_{e} / *v*_{p} = 5.93 × 10^{7} / 4.38 × 10^{6} = 13.54 : 1