around the sun, hence the 90-degree angle between the direction
of motion and the incoming sunlight. We assume that the sail is
100% reflective. The orientation of the sail is specified using the
angleθbetween the incoming rays of sunlight and the perpen-
dicular to the sail. In other words,θ=0 if the sail is catching the
sunlight full-on, whileθ=90◦means that the sail is edge-on to the
sun.
Conservation of momentum gives
pl i ght,i=pl i ght,f+∆psai l,where∆psai lis the change in momentum picked up by the sail.
Breaking this down into components, we have
0 =pl i ght,f,x+∆psai l,x and
pl i ght,i,y=pl i ght,f,y+∆psai l,y.As in example 53 on page 193, the component of the force that
is directly away from the sun (up in figure v/2) doesn’t change the
energy of the craft, so we only care about∆psai l,x, which equals
−pl i ght,f,x. The outgoing light ray forms an angle of 2θwith the
negativeyaxis, or 270◦− 2 θmeasured counterclockwise from the
xaxis, so the useful thrust depends on−cos(270◦− 2 θ) = sin 2θ.
However, this is all assuming a given amount of light strikes the
sail. During a certain time period, the amount of sunlight striking
the sail depends on the cross-sectional area the sail presents to
the sun, which is proportional to cosθ. Forθ=90◦, cosθequals
zero, since the sail is edge-on to the sun.
Putting together these two factors, the useful thrust is proportional
to sin 2θcosθ, and this quantity is maximized forθ≈ 35 ◦. A coun-
terintuitive fact about this maneuver is that as the spacecraft spi-
rals outward, its total energy (kinetic plus gravitational) increases,
but its kinetic energy actually decreases!
A layback example 71
The figure shows a rock climber using a technique called a lay-
back. He can make the normal forcesFN 1 andFN 2 large, which
has the side-effect of increasing the frictional forcesFF 1 andFF 2 ,
so that he doesn’t slip down due to the gravitational (weight) force
FW. The purpose of the problem is not to analyze all of this in de-
tail, but simply to practice finding the components of the forces
based on their magnitudes. To keep the notation simple, let’s
writeFN 1 for|FN 1 |, etc. The crack overhangs by a small, positive
angleθ≈ 9 ◦.
In this example, we determine thexcomponent ofFN 1. The other
nine components are left as an exercise to the reader (problem
81, p. 239).
Section 3.4 Motion in three dimensions 209