Re: position vs displacement (was Swartz letter in AJP (work-energy theorem))

[Hugh Logan <hloganPHY@CFL.RR.COM>]

Larry Smith wrote:

> At 5:56 PM -0400 9/2/04, Hugh Logan wrote to PHYS-L:
>
>
> > The use of "displacement" to label the axis of ordinates seems to have
> > popped up out of nowhere. At the very beginning of Chap. 3, they talk
> > about a position vs. time graph with the axes labeled "d" and "t."
> > However, they state that "d is used to represent its distance from the
> > origin." All their examples are consistent with d being a distance --
> > never negative. However (delta d) is described as a displacement as
> > previously mentioned, and it could be negative in the case of a downward
> > slope.
> >
> > In 1997-1998, I reviewed and corrected a number of errors, mostly
> > careless numerical errors, in fun@learning.physics....
> > one finds that the author, Dr. Mark
> > Sutherland, labels the axis of ordinates "displacement" rather than
> > "position"  on an x vs. t graph. In the text, he explicitly calls x
> > "displacement." I suggested that he call x position and delta x
> > "displacement," etc.
>
>
>
> > If
> > forced to think about it,  I always thought of a position vector as a
> > displacement from the origin, but it seemed inconvenient to think of a
> > displacement as a change of displacement, although that is what it
> > really is.
>
>
> I have struggled with this terminology (and its inconsistent use in the
> textbooks) for years; I think we even had a round of discussion on this
> list about it.
>
> Randy Knight's new book (PSE, 1e) is the first that takes a consistent
> approach, IMHO.  He does as you, Hugh, suggest: x is position and \Delta x
> is displacement.  At least this year I have a consistent story for my
> students rather than apologizing for the author's own confusion in previous
> texts.  But I don't know if I'm 100% totally happy yet, because I also
> agree with the strangeness indicated in your last paragraph.
Please don't give me credit for using x for position and Delta x for
displacement. I learned this from _PSSC Physics_ 2nd ed. back in 1969.
It remained through the seventh and final edition. I have noted on
several previous occasions that the good done by PSSC seems to have been
largely forgotten or ignored. I think the 7th ed. is still the best high
school text, even if one has to find a used copy.

> Furthermore, while there is a name for \Delta x (i.e. displacement), there
> isn't a corresponding one-word name for \Delta v.
>
> Knight uses the three stacked graphs and labels them position, velocity,
> and acceleration.  The derivative of the position graph is the velocity
> graph, but the area under the velocity curve (between t_i and t_f) is the
> "total displacement".  This is actually correct since x_f = x_i + the area
> under the curve, so moving the x_i over gives \Delta x = displacement =
> area under the curve; but I think it is a subtle point that is lost on most
> students.  They want the area under the curve to be the integral and they
> want the integral to be the inverse operation of the derivative, so they
> want the area under the curve to be the position, not the displacement.
If one uses Dr. Mark Sutherland's  idea of letting x be displacement,
then  Delta x is  change of displacement  (which is a displacement,
although one word doesn't characterize it very well.)  However,
it would satisfy the students' desire to have the result of the integral
of the velocity to be the inverse operation of the derivative, namely
the displacement. In this case, all three terms of

"x_f = x_i + the area under the curve"

would be displacement. I might have a hard time getting used to it. Many introductory texts including _PSSC Physics_, treat the area under the v vs. t graph as displacement. The latter introduces very few definitions and formulas
beyond displacement=Delta x, v(av)=(Delta x)/(Delta t),
a(av)=(Delta v)/(Delta t), and a=(Delta v)/(Delta t); also the instantaneous versions of v and a, even though not a calculus-based text. The students are forced to think graphically. For example, in the case of uniformly accelerated motion, Larry's "area under the curve" would be the area of a rectangle, (v_i)*(Delta t),  plus the area of a triangle, (1/2)*(Delta t)*[a*(Delta t)].
If t_i=0, (Delta t)=t_f=t, and one gets x_f=x_i+(v_i)*t+(1/2)*a*t^2 using
Larry's subscripts. Even without calculus, a graph of v vs. t makes it easy to see what the average velocity is: the constant velocity that would give the same area under the graph over the same time interval. In the case of uniformly accelerated motion, the areas of the right triangles above and below
the graph over the chosen time interval (above the rectangle if v is positive)
are of equal area (to make the total area correspond to the area under the average velocity) and hence congruent since the base, Delta t, is the same for
each. Then it is easy to see that v_av is midway between v_i and v_f or
v-av=(v_i+v_f)/2 for uniformly accelerated motion, but not necessarily so otherwise. A teacher in an algebra-based course could derive or lead the students to derive the formulas for uniformly accelerated motion once they had learned the material graphically. Not all of what I wrote was in PSSC. Lehrman and Swartz do about the same thing, but as noted, they prefer to work with the scalar quantities d=distance from origin and v=speed. Their v vs t graphs never cross the time axis, so their area is distance. PSSC gives the area beneath the time axis on a velocity (also v)vs. t graph a negative value. I think this is helpful to students going on to calculus or calculus-based physics.

One of the few calculus-based texts that I used in teaching, _University Physics_, 7th ed., by Sears, Zemansky, and Young uses essentially the same notation as that of Randy Knight, except that the subscripts are "1" and "2" instead of "i" and "f." They describe the area under the v vs. t graph as displacement, writing (on p. 35),

x_2-x_1= (integral from x_1 to x_2 of dx)=(integral from v_1 to v_2 of v*dt).

Likewise,

v_2-v_1= (integral from v_1 to v_2 of dv)=(integral from t_1 to t_2 of a*dt).

Similarly, _Elementary Classical Physics_ by Weidner and Sells, 1973 that I used for AP Physics in 1976-1977, uses exactly the same notation as Randy Knight, even the subscripts "i" and "f." However, for uniformly accelerated motion problems, they simplify it by choosing t_i=t_0=0, using the subscript "0" for initial values and no subscript for final values so that
x=x_0+(v_0)*t+ (1/2)*a*t^2. They also show the graphical interpretation of
average velocity that I described above. This was a good, accurate, no-nonsense book, but it was probably not glitzy enough to survive.

The Giancoli texts that I have used or seen including _Physics_3rd, 5th, and 6th editions all use about the same notation as Sears, Zemansky, and Young, switching to subscript "0" for initial and no subscript for final in the case of uniformly accelerated motion. Similarly for _Physics for Scientists and Engineers_2nd ed., which was used as a supplement for about four calculus-based students in the course based on _Physics_, 5th ed. They also use subscript "i" on the (Delta t) in the Riemann sums. PSE was a considerably
more advanced text than all the others mentioned.

Once one is used to the notation of these books, it is difficult to go back to those that essentially plot distance vs. time, and use speed instead of velocity. This might be OK for ninth grade physics, which I wouldn't want to teach. Hewitt's _Conceptual Physics_, 5th ed. is very qualitative. He deals mostly with distance and speed, writing equations out in words rather than symbols at first. On p. 25 of the 8th ed., they state, "Loosely speaking, we can use the words _speed_ and _velocity_ interchangeably," but make the distinction that velocity has direction. On p. 27, they write

"Acceleration (along a straight line) = (change in speed)/(time interval)"

for the case "when the direction is not changing." I presume they mean for uniform acceleration. This would lead to results that disagree with

Acceleration (along a straight line) = (change in velocity)/(time interval)

where the direction of velocity is given as a signed quantity in the case that the motion can be in the negative direction, which is probably intended to be excluded. According to Hewitt's definition, a body always has negative acceleration when it is slowing down. As they used to say, "Deceleration is negative acceleration." I suppose this is OK if the positive direction is implicitly chosen as the direction of motion, but it is not useful in cases where the direction changes as in the case of a ball thrown upwards. Hewitt does give a correct definition of acceleration in words in the "Summary of Terms" on p.32 of the 8th ed.

I suppose these are small considerations for teachers, especially those not
trained primarily as physicists (and many that are), who want to see "physics
first" as a background for chemistry and biology.

I hope there will always be a little room for physics at a higher grade level, as _PSSC Physics_ was for senior level physics. John Clement often mentions _Minds-On Physics_. I have seen only a few samples on the UMPERG web site, and I don't know at what grade level it is primarily used. I suspect Modeling Instruction is attentive to the mathematics involved in physics as a representational tool. I gather it is used at a variety of grade levels, but I have no experience with it.

I thought Larry Smith was going to catch me on the number of dimensions that I used on my previous post -- as he did once before. I thought of it later.


> Dr. Mark Sutherland, labels the axis of ordinates "displacement"
> rather than "position"  on an x vs. t graph. In the text, he
> explicitly calls x "displacement." I suggested that he call x position
> and delta x "displacement," etc. This was one suggestion he did not
> accept. I no longer have his reply, but as I recall, he noted that if
> one parallel transports a displacement (in two or higher dimensions),
> it is still equal to the original displacement. If one translates a
> position vector, the descriptor, "position," would not pertain to the
> same origin.

That should be one or higher dimensions, as one is a degenerate case. The displacement, thought of as an arrow, can be slid along the line of motion
without changing its value. This reminds me of the way numbers and differences of numbers were treated as arrows when discussing the number line in ninth grade algebra -- if memory serves me correctly.

Hugh
Flagler Beach, FL
[Where one thinks about physics as hurricanes approach.]