91Ó°ÊÓ

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 \(9.0 \times {10^5}\;{\rm{m/s}}\).