I have found that one dimensional kinematics/dynamics is clarified for
the
un-washed if every one dimensional vector is explicitly written as a
magnitude and a unit vector. One need only define, once and for all,
the
positive direction of a single unit vector.
For example, in vertical free fall, define [k] as the unit vector
pointing vertically
up, once and for all. Then the free fall acceleration vector is
always -g[k], where g is the positive magnitude 32 ft/s^2 or 9.8
m/s^2.
The vertical free fall velocity vector is then given by:
v[k] = vo[k] -tg[k] . Stress that the numbers v and vo carry a sign,
and
[k] always denotes the same (upward) direction.
After students are used to this formalism, they can "cancel" the unit
vector out of such one dimensional vector equations. I think it
should be
stressed that this is only to ease writing pains; the unit vector and
its defined direction is still
implicitly there - in each term. They are one-dimensional VECTOR
equations.
PS: All of the above could be done using the alternate definition of
[k]
as vertically DOWN.
Then the free fall acceleration is always g[k], with g=32ft/s^2 . . .
and
v[k] = vo[k] + tg[k]
In assigning/interpreting numerical values to/of v and vo, due account
must be taken of one's previous choice of definition for the unit
vector
[k].
PPS: This pedagogy is why I always introduced the general 3D vector
concept and the unit vector concept BEFORE doing one-dimensional
kinematics.
Bob Sciamanda
Physics, Edinboro Univ of PA (Emeritus)
www.winbeam.com/~trebor
trebor@winbeam.com