Light of wavelength `lamda` from a small 0.5 mW He-Ne laser source, used in the school laboratory, shines from a spacecraft of mass 1000 kg. Estimate the time needed for the spacecraft to reach a velocity of `1.0km^(-1)` from rest. The momentum p of a photon of wavelength `lamda` is given by `p=(h)/(lamda)`, where h is Planck's constant.

Options

A. `6..10^(18)`
B. `3xx10^(17)`
C. `6xx10^(17)`
D. `2xx10^(15)`

Answer

C

Explanation

Photons have momentum `(P=(h)/(lamda))` which they carry away, the spacecraft will acquire momentum in the opposite direction according to law of conservation of momentum.
No. of photons per second from laser =n
Then, from energy consideration.
`0.5xx10^(-3)=nh((c )/(lamda))`
`n=(0.5xx10^(-3))(lamda)/((ch))`
Rate of change of momentum of spacecraft
`np=n(h)/(lamda)=(0.5xx10^(-3))(lamda)/(ch)((h)/(lamda))=(0.5xx10^(-3))/(c )`
From Newton's second law, `(nh)/(lamda)=ma`
`1000a=(0.5xx10^(-3))/(3.00xx10^(8))=(1)/(6)xx10^(-11)`
`v=at`
`t=(v)/(a)=(1000)/((1)/(1000)xx(1)/(6)xx10^(-11))s=6xx10^(17)s`