Simple Nature - Light and Matter

ai/An object moves through

a field of force.

have

K=

∑n

j=1

1

2

mj(vj+u)·(vj+u)

=

1

2





∑n

j=1

mjvj·vj+ 2

∑n

j=1

mjvj·u+

∑n

j=1

mju·u



.

As in the proof on page 132, the first sum is simply the total kinetic
energy in the original frame of reference, and the last sum is a con-
stant, which has no effect on the validity of the conservation law.
The middle sum can be rewritten as

2

∑n

j=1

mjvj·u= 2u·

∑n

j=1

mjvj

= 2u·

∑n

j=1

pj,

so the only way energy can be conserved for all values ofuis if the

vector sum of the momenta is conserved as well.

3.4.6 Gradients and line integrals (optional)
This subsection introduces a little bit of vector calculus. It can
be omitted without loss of continuity, but the techniques will be
needed in our study of electricity and magnetism, and it may be
helpful to be exposed to them in easy-to-visualize mechanical con-
texts before applying them to invisible electrical and magnetic phe-
nomena.
In physics we often deal with fields of force, meaning situations
where the force on an object depends on its position. For instance,
figure ai could represent a map of the trade winds affecting a sailing
ship, or a chart of the gravitational forces experienced by a space
probe entering a double-star system. An object moving under the
influence of this force will not necessarily be moving in the same
direction as the force at every moment. The sailing ship can tack
against the wind, due to the force from the water on the keel. The
space probe, if it entered from the top of the diagram at high speed,
would start to curve around to the right, but its inertia would carry
it forward, and it wouldn’t instantly swerve to match the direction
of the gravitational force. For convenience, we’ve defined the gravi-
tational field,g, as the forceper unit mass, but that trick only leads
to a simplification because the gravitational force on an object is
proportional to its mass. Since this subsection is meant to apply
to any kind of force, we’ll discuss everything in terms of the actual
force vector,F, in units of newtons.
If an object moves through the field of force along some curved
path from pointr 1 to pointr 2 , the force will do a certain amount

Section 3.4 Motion in three dimensions 219