Download Historical burdens on physics 57 Instantaneous and average velocity

Historical burdens on physics
57 Instantaneous and average velocity
Subject:
In a physics school book I found the following highlighted statements:
“By velocity v of a uniform motion we understand the constant quotient of
an arbitrary displacement Δs and the time Δt necessary for this displacement:
v = Δs/Δt .”
“For a uniform movement with initial values t = 0 and s = 0 in addition to v =
Δs/Δt there is: v = s/t .“
“In reality we find that the instantaneous velocity is approximately equal to
the average velocity in a time interval that is as small as possible.”
And in another text book, highlighted as well:
“Definition: If for a rectilinear movement of a body the displacement s and
the time t are proportional to one another, the constant quotient s/t = v is
called the velocity of the body.”
“Definition: If in a section of a rectilinear movement all the quotients Δs/Δt
have the same value, then Δs/Δt = v is the velocity within this section.”
“If Δs and Δt are intervals of the displacement and the time of an arbitrary
movement, which belong to one another, then
Δs
v =
Δt
is the average velocity of this movement over the displacement Δs or the
interval of time Δt respectively.”
“The instantaneous velocity at time t0 is obtained approximately by taking
the interval velocity of a time interval which is as small as possible and
which contains t0.”
Similar propositions are found for the acceleration.
Such statements are not a peculiarity of those books from which they are
taken. They can be found in many other physics text books, whether they
are new and have a modern outfit or they are a hundred years old.
Deficiencies:
The concept of velocity is introduced with an unusual meticulousness. Several concerns may arise.
1. The rigorousness cannot be maintained subsequently. It is to compare
with the looseness of the introduction of the concepts force, heat or electric
current intensity.
2. Just at the beginning of the physics course such a formalization has a
daunting effect on the students.
3. The distance is not great between rigorous thinking to pedantry. One may
ask if in the present case the limit to pedantry has not been crossed.
4. It is said that velocity is defined by Δs/Δt. It is not said that v = Δs/Δt is
the relation between v, s and t. Do we want to say to the students that the
concept of velocity which they had before is not the velocity in the sense of
physics? We should better not present a mathematically embellished triviality as a new insight. By the way: If one insists to define velocity, this can be
done yet in another way.1
5. What is offered as a way to understand velocity is not really handy. The
detour leads over two or three special velocities: the instantaneous, the interval and the average velocity. If with other physical quantities we would
proceed in a similar way we would not get very far. Here is what we had to
claim when introducing other quantities: “In reality we find that the instantaneous electric current intensity is approximately equal to the average intensity in a time interval that is as small as possible.” Or: “The local mass density is approximately equal to the average density in a region of space that
is as small as possible.”
Origin:
Probably a legacy from the beginnings of physics as a science. In text
books from the 18th century one meets a similar meticulousness also in
other contexts, where we nowadays do not see any problem.
Disposal:
A disarmament can be reached in several ways. It is not necessary to explain what is meant by velocity and what is meant by constant velocity. The
equation v = s/t describes the relation between the velocity, the travelled
distance and the time that is needed in the case that the velocity is constant. If it is not constant, we proceed in the same way as we do with other
physical quantities whose values changes with time. The velocity is measured with the tachometer.
1
We sketch two other procedures for defining velocity. However, we do not
recommend to use these definitions at school.
1. Velocity can be defined by means of the relation:
dE = vdp, i.e. energy change per momentum change. This is analogous to
the definition of the electric potential difference (energy change per change
of electric charge) or that of the absolute temperature (energy change per
entropy change).
2. Velocity can be defined operationally. By means of tachometer which
must not be calibrated, we can ascertain if a velocity is constant in time. So
we can define a unit v0 of the velocity. Now, multiples of this unit can easily
be constructed. A body A is moved with velocity v0 relative to a body B,
which for his part moves with velocity v0 relative to the earth. Now A has the
velocity 2 v0 relative to the earth.
Friedrich Herrmann, Karlsruhe Institute of Technology