How to generalize one-dimensional equations example 66
.How can the one-dimensional relationshipspt ot al=mt ot alvcmandxcm=∑
∑jmjxj
jmj
be generalized to three dimensions?
.Momentum and velocity are vectors, since they have directions
in space. Mass is a scalar. If we rewrite the first equation to show
the appropriate quantities notated as vectors,pt ot al=mt ot alvcm,we get a valid mathematical operation, the multiplication of a vec-
tor by a scalar. Similarly, the second equation becomesrcm=∑
∑jmjrj
jmj,
which is also valid. Each term in the sum on top contains a vector
multiplied by a scalar, which gives a vector. Adding up all these
vectors gives a vector, and dividing by the scalar sum on the bot-
tom gives another vector.
This kind of wave-the-magic-wand-and-write-it-all-in-bold-face tech-
nique will always give the right generalization from one dimension
to three, provided that the result makes sense mathematically —
if you find yourself doing something nonsensical, such as adding
a scalar to a vector, then you haven’t found the generalization
correctly.
Colliding coins example 67
.Take two identical coins, put one down on a piece of paper, and
slide the other across the paper, shooting it fairly rapidly so that
it hits the target coin off-center. If you trace the initial and final
positions of the coins, you can determine the directions of their
momentum vectors after the collision. The angle between these
vectors is always fairly close to, but a little less than, 90 degrees.
Why is this?
.Let the velocity vector of the incoming coin bea, and let the two
outgoing velocity vectors bebandc. Since the masses are the
same, conservation of momentum amounts toa=b+c, which
means that it has to be possible to assemble the three vectors
into a triangle. If we assume that no energy is converted into
heat and sound, then conservation of energy gives (discarding
the common factor ofm/2)a^2 =b^2 +c^2 for the magnitudes of the206 Chapter 3 Conservation of Momentum