Calculating Force Exerted by Photons on a Small Space Vehicle

[Amith2006]

Homework Statement

It is desired to move a small 50 kg space vehicle by a lamp which emits 100 watts of blue light(lambda= 4700 Angstrom). If the vehicle is in free space, what will its acceleration?

Homework Equations

The Attempt at a Solution

The solution as given in my book is as follows:

Here we take the particle nature of light. Since the emitted light carries momentum the change in momentum per second will be responsible for exerting force on the vehicle.

Suppose the lamp emits N photons per second. Then,
Nhf = Power of lamp
Where h = Planck’s constant, f = frequency of light
On simplification we get,
N= 2.4 x 10^20
Each photon will have a momentum given by,
p = h/lambda
= 1.4 x 10^(-27) N – sec
The total force on the vehicle as per Newton’s 2nd law is,
F = d/dt(Np)
= p(dn/dt)
= 3.4 x 10^(-7) N
Therefore,
acceleration = F/mass
= 6.8 x 10^(-9) m/sec^2
I understood till the momentum calculation. I didn’t understand the force calculation. I calculated it in the following way:
The change in momentum of each photon after striking the space vehicle per second= 2p
change in momentum of N photons per second = 2pN
F = dp/dt
= 2pN
= 3.32 x 10^(-6) N

Could somebody please guide me?

 

[Kurdt]

Well what you have done is tried to calculate the momentum change of the photon when what you want is the momentum imparted to the ship by the photons.

 

[OlderDan]

Could somebody please guide me?

I don't see how you got your number from computing 2pN. If you use their values of p and N, you should get twice their result. They apparently assumed the photons are being absorbed, while you are assuming they are being reflected, so your force is twice as much as theirs. That will give you a factor of 2 difference between your result and theirs.

 

[Amith2006]

You are right. I never thought about it. Thanks.