# Would a laser gun recoil?

Today I'd like to approach a question near and dear to many a geek
heart: do laser guns have recoil?
Let's motivate our question a little bit. I've wondered about this
question since I saw star wars. Though I'm no firearms expert, the
recoil in guns must come from conservation of momentum principles.
Momentum is conserved in a system. The gun starts with zero momentum. We
fire, give the bullet momentum, and so to keep the system at zero
momentum, the gun must gain equal and opposite momentum. That is, the
gun will move backwards. All of that was for conventional guns. Light
carries momentum, so if we fire a pulse of light, we expect our laser
gun to recoil. So yes, they do have recoil. Satisfied, dear readers?
Neither am I. The question we really mean to ask is, does a laser gun
have noticeable recoil? We need to make a few reasonable assumptions.
Let's assume that the laser gun fires a pulse with as much energy as a
bullet has kinetic energy, KE. The energy, E, of light is related to its
momentum, p, by E=pc, where c is the speed of light. This gives a
momentum of $$ E=KE=pc $$ $$ p=\frac{KE}{c}$$ What is the kinetic
energy of a bullet? A little searching reveals that a .22 bullet is
\~2.5g and fires with a muzzle velocity of \~330m/s. Kinetic energy is
given by KE=1/2mv^2, where m is mass and v velocity. So, the momentum
of a laser pulse with equal energy would be $$ p=\frac{mv^2}{2c} $$ $$
p=\frac{.0025kg*(330m/s)^2}{2*3\cdot 10^8m/s} $$ $$
p=4.5\cdot10^{-7}kg \cdot m/s $$ For comparison, the momentum (p=mv)
of a .22 bullet is .83 kg*m/s. The momentum of a laser gun is 2 million
times less than the momentum of a .22. But is momentum all we should
consider? I suspect the 'kick' we feel on the recoil is directly related
to the force that the gun exerts on the holder. This means that instead
of momentum we need to consider impulse, momentum per time. We estimate
the time it takes to fire a .22 is \~.1s, so the force delivered 8.3 N.
Let's estimate the time it takes a laser gun to fire. Unfortunately, not
having a laser gun to fire (feel free to send me one, dear readers),
we're more or less going to have to guess at the firing time. Most
movies with laser guns show pulses of light (which, incidentally would
move so fast we wouldn't see them) on the order of a meter or two long.
Given the speed of light, this would give a firing time of \~30
nanoseconds. This would give a force delivered of 15 N. This is close to
what we estimated or a .22. So, if movies are to be believed (and
really, why wouldn't we believe them?), it seems like laser guns may
well have recoil. *Note: It is worth questioning if we need the same
energy for a laser as for a bullet. That could certainly change our
estimate. Maybe we'll return to this question again.*

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