Calculate the change in kinetic energy of an electron when the velocity changes by a small amount.
Consider an electron traveling at a velocity of 10^{7} \mathrm{~cm} / \mathrm{s}. Assume that the velocity increases by a value of 1 \mathrm{~cm} / \mathrm{s}. The increase in kinetic energy is given by
\Delta E=\frac{1}{2} m v_{2}^{2}-\frac{1}{2} m v_{1}^{2}=\frac{1}{2} m\left(v_{2}^{2}-v_{1}^{2}\right)
Let v_{2}=v_{1}+\Delta v. Then
v_{2}^{2}=\left(v_{1}+\Delta v\right)^{2}=v_{1}^{2}+2 v_{1} \Delta v+(\Delta v)^{2}
But \Delta v \ll v_{1}, so we have that
\Delta E \approx \frac{1}{2} m\left(2 v_{1} \Delta v\right)=m v_{1} \Delta v
