d/Halley’s comet. Top: A
photograph made from earth.
Bottom: A view of the nucleus
from the Giotto space probe.(W.
Liller and European Space Agency)all the exhaust has been expelled. If we define the positive di-
rection as the direction the spacecraft is going, then the negative
momentum of the exhaust is canceled by the positive momen-
tum of the spacecraft. The ion drive allows a final speed that is
ten times greater. (This simplified analysis ignores the fact that
the reaction mass expelled later in the burn is not moving back-
ward as fast, because of the forward speed of the already-moving
spacecraft.)3.1.2 Nonmechanical momentum
So far, it sounds as though conservation of momentum can be
proved mathematically, unlike conservation of mass and energy, which
are entirely based on observations. The proof, however, was only for
a mechanical system, with interactions of the formU(r). Conser-
vation of momentum can be extended to other systems as well, but
this generalization is based on experiments, not mathematical proof.
Light is the most important example of momentum that doesn’t
equalmv— light doesn’t have mass at all, but it does have mo-
mentum. For example, a flashlight left on for an hour would absorb
about 10−^5 kg·m/s of momentum as it recoiled.
Halley’s comet example 4
Momentum is not always equal tomv. Halley’s comet, shown in
figure d, has a very elongated elliptical orbit, like those of many
other comets. About once per century, its orbit brings it close
to the sun. The comet’s head, or nucleus, is composed of dirty
ice, so the energy deposited by the intense sunlight gradually re-
moves ice from the surface and turns it into water vapor. The
bottom photo shows a view of the water coming off of the nucleus
from the European Giotto space probe, which passed within 596
km of the comet’s head on March 13, 1986.
The sunlight does not just carry energy, however. It also carries
momentum. Once the steam comes off, the momentum of the
sunlight impacting on it pushes it away from the sun, forming a
tail as shown in in the top image. The tail always points away
from the sun, so when the comet is receding from the sun, the tail
is in front. By analogy with matter, for which momentum equals
mv, you would expect that massless light would have zero mo-
mentum, but the equationp=mvis not the correct one for light,
and light does have momentum. (Some comets also have a sec-
ond tail, which is propelled by electrical forces rather than by the
momentum of sunlight.)
The reason for bringing this up is not so that you can plug
numbers into formulas in these exotic situations. The point is that
the conservation laws have proven so sturdy exactly because they
can easily be amended to fit new circumstances. The momentum of
light will be a natural consequence of the discussion of the theory
of relativity in chapter 7.
Section 3.1 Momentum in one dimension 135