Some scientists have suggested that spacecraft with sails of the kind described

Question

Some scientists have suggested that spacecraft with sails of the
kind described in Conceptual Example 4 could be propelled by lasers.
Suppose that such a sail is constructed of a highly reflective material
thin enough so that one square meter of the sail has a mass of just
3.0 103 kg. The sail is to be propelled by an ultraviolet laser beam
(wavelength 225 nm) that will strike its surface perpendicularly.
(a) Use the impulse–momentum theorem (Section 7.1) to determine
the number of photons per second that must strike each square
meter of the sail in order to cause an acceleration of 9.8 106 m/s2,
which is one million times smaller than the gravitational acceleration
at the earth’s surface. Assume that no other forces act on the sail,
and that all the incident photons are reflected. (b) Determine the
intensity (power per unit area) that the laser beam must have when it
strikes the sail.

Explanation / Answer

a)

here

dP = F * dT

F = dP / dT = m * dV / dT = rho * A * dV/dT

the chane in momentum of a single photon dp = 2*p = 2 * h / L

the number of photons N = (rho * A * dV/dt ) / dp

N / A = ( 3 * 10^3 * 9.8 * 10^6 * 225 * 10^-9) / (6.63 * 10^-34)

N/A = 9.97 * 10^36 photons / sm^2

b)

the energy per photon

E = h * F = h * c / L

multiply this by N/A

= (N/A) * h * c / L

the intensity = (6.63 * 10^-34 * 9.97 * 10^36 * 3 * 10^8) / (225 * 10^-9)

= 8.813 * 10^18 J/s

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