If the acceleration is constant, then the average
velocity during a time interval (t1 < t < t2)
equals the instantaneous velocity at the midpoint
of the time interval [t = (t1 + t2)/2]. If the
midpoint of the time interval is used with the
data given by Ludwik, then the calculated
acceleration is inconsistent with the assumption
that it is constant.
I obtain the following values:
t (s) 0.75 2.0 2.9 3.55
Vav (m/s) 1.33 2.0 2.5 4.0
Aav (m/s^2) 0.536 0.556 2.31
The first two calculated values of acceleration
are consistent with the assumption that the
acceleration is constant, but the third value
is clearly inconsistent with that assumption.
If the acceleration is increasing (decreasing)
monotonically, then the average velocity equals
the instantaneous velocity at a time later
(earlier) than the midpoint of the interval.
If the final time reading was 4.0 s, then the
final average velocity would be 2.86 m/s and
would correspond to the instantaneous velocity
at t = 3.65 s, assuming that the acceleration
was constant. In that case, the acceleration
for the final interval would be 0.514 m/s^2,
which would be more consistent with the
assumption of constant acceleration.
Having students run provides a kinesthetic
experience that can aid learning, but it is
difficult for a runner to maintain a constant
acceleration for an extended time.
We perform a somewhat similar experiment
using a cart accelerated across a table by a
hanging mass. We use a spark timer to record
the displacement of cart every 0.1 s. This
experiment provides an acceleration that is
essentially constant, but does not provide
a kinesthetic experience.
Daniel Crowe
Oklahoma School of Science and Mathematics
Ardmore Regional Center
dcrowe@sotc.org
-----Original Message-----
From: Ludwik Kowalski [mailto:kowalskil@MAIL.MONTCLAIR.EDU]
Sent: Saturday, September 13, 2003 7:44 PM
To: PHYS-L@lists.nau.edu
Subject: Kinematics
I have questions based on the first lab this year.
I took students outdoor. With chalk we marked
positions along a path (at x=0, 2, 4, 6, etc. meters
from the origine. Two students at each mark
recorded time instances at which a runner was
passing by; all stopwatches were pressed at the
same moment, that is when x was zero. Here are
the data I am inventing to ask my questions:
1 x (m) 0 2 4 6 8
2 t (s) 0 1.5 2.5 3.3 3.8
3 dx 2 2 2 2
4 dt 1.5 1.0 0.8 0.5
5 Vav 1.33 2.0 2.5 4.0
6 ACav A B C
SPACES IN THIS TABLE SHOULD BE
PRESERVED IF YOUR FONT IS COURIER.
The goal is to calculate average accelerations,
A, B and C. Also to plot Vav and ACav versus time.
Lines 1 and 2 show row data (positions and times).
Lines 3 and 4 show steps in space and in time
Line 5 shows average velocities (dx/dt).
Question #1
In plotting Vav versus time what value of t should be
associated with each Vav? For example, should it
be (1.5+2.5)/2=2 s for Vav=2.0 and 2.6 for Vav=2.5?
Question #2
What value of dt should be used to calculate average
accelerations? For example, what should dt be in
B=(2.5-2.0)/dt ? Should it be 0.9?
Note: the concept of instantenous v and a has not
been introduced. dx and dt are deltas.
Ludwik Kowalski