Quite apart from effects due to Earth’s rotational and orbital motions, a laboratory reference frame is not strictly an inertial frame because a particle at rest there will not, in general, remain at rest; it will fall. Often, however, events happen so quickly that we can ignore the gravitational acceleration and treat the frame as inertial. Consider, for example, an electron of speed v =0.992c, projected horizontally into a laboratory test chamber and moving through a distance of 20 cm. (a) How long would that take, and (b) how far would the electron fall during this interval? (c) What can you conclude about the suitability of the laboratory as an inertial frame in this case?

The speed of electron is v = 0.992 c

The distance travelled by electron in chamber is d = 0.20 cm

The wavelength of light for observer on Earth is calculated by using the formula for Doppler effect in light and color of light is decided by wavelength of light.

a) The time taken by electron to cover distance is given as:

$t=\frac{d}{v}$

Here, c is the speed of light and its value is $3\times {10}^{8}m{s}^{-1}$.

Substitute all the values in the above equation.

$t=\frac{\left(20cm\right)\left({\displaystyle \frac{1cm}{100cm}}\right)}{0.992c\left({\displaystyle \frac{3\times {10}^{8}m{s}^{-1}}{c}}\right)}$

$t=6.72\times {10}^{-10}s$

Therefore, the time taken by electron to cover distance is $6.72\times {10}^{-10}s$.

b) The distance fallen by electron in chamber is given as:

$y=\frac{1}{2}{g}^2$

Here, g is the gravitational acceleration and its value is $9.8m{s}^{-2}$.

Substitute all the values in the above equation.

$$\begin{gathered} y=\frac{1}{2}\left(9.8\text{m}/{\text{s}}^2\right){\left(6.72\times {10}^{-10}\text{s}\right)}^2 \\ =2.213\times {10}^{-18}\text{m} \end{gathered}$$

Therefore, the distance fallen by electron in chamber is $2.213\times {10}^{-18}m$.

c)

As the distance fallen by the electron is very less compared to horizontal distance of electron in chamber due to negligible effect of gravity on electron and very high speed of electron, so the laboratory can be considered as inertial frame for experiment.

Therefore, the laboratory can be considered as inertial frame for experiment.