Question

Suppose you designed a spacecraft to work by photon pressure. The sail was a completely absorbing fabric of area 1.0 km2 and you directed a laser beam of wavelength 650 nm onto it at a rate of 1 mol of photons per second from a base on the moon. What is
(a) the force,
(b) the pressure exerted by the radiation on the sail?
(c) The spacecraft has a mass of 1.0 kg. Given that, after a period of acceleration from standstill, speed = (force/mass) x time, how many minutes would it take for the craft to accelerate to a speed of 1.0 m/s (about 2.2 mph)?

Answer 1

Area of sail $A=1.0\,km^2=1.0*10^{6}\,m^2$

Wavelength of light $\lambda=650\,nm=650*10^{-9}\,m$

Rate of impact of photons $R=1\,mol/s=6.022*10^{23}\,photons/s$

(a)

Momentum of each photon is $p'=\frac{h}{\lambda}=\frac{6.625*10^{-34}}{650*10^{-9}}=1.0192*10^{-27}\,kg.m/s$

Since the photons are absorbed completely, in every collision the above momentum is transferred to the sail.

Momentum transferred to the sail per second is product of rate of impact of photons and momentum transferred by each photon.

$\frac{dp}{dt}=R*\frac{h}{\lambda}$

This is the force acting on the sail.

$F=R*\frac{h}{\lambda}=6.022*10^{23}*1.0192*10^{-27}=6.14*10^{-4}\,N$

b)

Pressure exerted by the radiation on the sail = Force acting on the sail / Area of the sail

$P=\frac{F}{A}=\frac{6.14*10^{-4}}{1.0*10^{6}}=6.14*10^{-10}\,Pa$

c)

Acceleration of spacecraft $a=\frac{F}{m}=\frac{6.14*10^{-4}}{1.0}=6.14*10^{-4}\,m/s^2$

As the spacecraft starts from rest, initial speed $u=0\,m/s$ , final speed is $v=1.0\,m/s$ after time $t$

$v=u+at$

$\Rightarrow t=\frac{v-u}{a}=\frac{1.0-0}{6.14*10^{-4}}=1629\,s=27.2\,min$