(II) What is the speed of a proton whose KE is 4.2 keV?

The kinetic energy of an object relies on the mass of the object and its speed.

The kinetic energy of an object is given by,

\(KE = \frac{1}{2}m{v^2}\) … (i)

Here, m is the mass and v is the speed.

The kinetic energy of the proton is, \(KE = 4.2\;{\rm{keV}}\)

Mass of the proton is, \(m = 1.67 \times {10^{ - 27}}\;{\rm{kg}}\)

The kinetic energy of the proton in joule is,

\(\begin{aligned}K{E_1} &= 4.2\;{\rm{keV}}\\ &= \left( {4.2 \times {{10}^3}\;{\rm{eV}}} \right) \times \left( {\frac{{1.6 \times {{10}^{ - 19}}\;{\rm{J}}}}{{1\;{\rm{eV}}}}} \right)\\ &= 6.72\; \times {10^{ - 16}}{\rm{J}}\end{aligned}\)

From equation (i), the speed of proton is calculated as:

\(v = \sqrt {\frac{{2\left( {KE} \right)}}{m}} \)

Substitute the values in the above expression.

\(\begin{aligned}{v_1} &= \sqrt {\frac{{2\left( {6.72 \times {{10}^{ - 16}}\;{\rm{J}}} \right)}}{{1.67 \times {{10}^{ - 27}}\;{\rm{kg}}}}} \\ &= \sqrt {80.4790 \times {{10}^{10}}} \;{\rm{m/s}}\\ &\approx 9.0 \times {10^5}\;{\rm{m/s}}\end{aligned}\)

Thus, the speed of the proton with kinetic energy 4.2keV is \(9.0 \times {10^5}\;{\rm{m/s}}\).