Light of wavelength $\lambda$ from a small $0.5mW He-Ne$ laser (source used in school laboratories) shines from a spacecraft of mass $1000kg$.  Estimate the time needed for the spacecraft to reach a velocity of $1.0km s^{-1}$ from rest. 

Photons have momentum $\left(p=\frac{h}{\lambda }\right)$ which they carry away; the spacecraft will acquire momentum in the opposite direction according to law of conservation of momentum.

No. of photons per sec. from laser $=n$.

Then from energy considerations,

$$\begin{gathered} 0.5\times {10}^{-3}=nh\left(\frac{c}{\lambda }\right) \\ n=\left(0.5\times {10}^{-3}\right)\lambda /\left(ch\right) \end{gathered}$$

Rate of change of momentum of spacecraft

$$\begin{gathered} =np=n\frac{h}{\lambda }=\left(0.5\times {10}^{-3}\right)\frac{\lambda }{ch}\left(\frac{h}{\lambda }\right) \\ =\frac{0.5\times {10}^{-3}}{c} \end{gathered}$$

From Newton's second law,

$ma=\frac{0.5\times {10}^{-3}}{c}$

or  $1000a=\frac{0.5\times {10}^{-3}}{3.00\times {10}^8}=\frac{1}{6}\times {10}^{-11}$

But $v=at$

So $t=\frac{v}{a}=\frac{1000}{\left({\displaystyle \frac{1}{1000}}\right)\times {\displaystyle \frac{1}{6}}\times {10}^{-11}}sec$

$=6\times {10}^{17}sec$