Simple Nature - Light and Matter

i/Example 57.

j/The geometric interpreta-

tion of a vector’s components.

be eliminated by dividing each equation bym, and we find

v 3 x=−1.0x10^4 km/hr

v 3 y=−1.0x10^4 km/hr,

which gives a magnitude of

|v 3 |=

√

v 32 x+v 32 y

= 1.4x10^4 km/hr.

A toppling box example 57

If you place a box on a frictionless surface, it will fall over with a

very complicated motion that is hard to predict in detail. We know,

however, that its center of mass’s motion is related to its momen-

tum, and the rate at which momentum is transferred is the force.

Moreover, we know that these relationships apply separately to

each component. Letxandybe horizontal, andzvertical. There

are two forces on the box, an upward force from the table and a

downward gravitational force. Since both of these are along the

zaxis,pzis the only component of the box’s momentum that can

change. We conclude that the center of mass travels vertically.

This is true even if the box bounces and tumbles. [Based on an

example by Kleppner and Kolenkow.]

Geometric representation of vectors
A vector in two dimensions can be easily visualized by drawing
an arrow whose length represents its magnitude and whose direction
represents its direction. Thexcomponent of a vector can then be
visualized, j, as the length of the shadow it would cast in a beam of
light projected onto thexaxis, and similarly for theycomponent.
Shadows with arrowheads pointing back against the direction of the
positive axis correspond to negative components.
In this type of diagram, the negative of a vector is the vector
with the same magnitude but in the opposite direction. Multiplying
a vector by a scalar is represented by lengthening the arrow by that
factor, and similarly for division.
self-check K
Given vectorQrepresented by an arrow below, draw arrows represent-
ing the vectors 1.5Qand−Q.

.Answer, p. 1056

Section 3.4 Motion in three dimensions 199