One Dimensional Motion
This lab has a dual purpose. We want you to become familiar with the operation of a
linear air track and recognize this as another instance of the cultural preference that
physicists have towards simplicity and minimalism. You will then use it to investigate
the motion of a body subjected to a constant force and discover the functional
relationship between distance, velocity, and time.
Required for this lab:
a) A good understanding of the lab reading material provided here.
b) Science Workshop – Data Studio program
c) KaleidaGraph or Graphical Analysis. So familiarize yourself with KaleidaGraph
or Graphical Analysis by reading the information available in the lab manual
reading material for “Introduction to Our Curve Fitting Computer Programs”
Prior reading and preparing for your experiment well ahead will save you a great deal
of time and frustration in the lab.
“Kinematics” is the study of motion, as opposed to “dynamics”, where one attempts to
understand the “cause” of motion. In this experiment you will analyze one-dimensional
motion and determine the equation of motion, expressed in the form:
x t = f (t) (1)
where xt is the location of the object at time t.
To study the one-dimensional motion, you will use a cart and an air track. An air track is
a nearly frictionless device, which can be used to study motion under ideal conditions.
Carefully inspect all parts of the air track and note their function. While nothing is
breakable during ordinary use, it is a delicate instrument and should be treated as such.
In particular, don’t drop the carts! If a cart is dropped, contact your instructor and ask
them to inspect it carefully. With the blower on, inspect all the air holes to ensure that
they are open.
In practice, we can never (or can we?) achieve a perfectly frictionless environment; even
a levitated cart set in motion will eventually come to rest. Thus, you should investigate
how good an approximation it is to consider the air track frictionless.
The simplest example of motion is an object moving at constant velocity, vo, in a straight
line. In this case, the change in an objects position after a period of time, t, is trivial.
"x = v o t (2)
When the object is accelerating (i.e. the air track is inclined, there is substantial friction,
there is a big draft of air, etc.) this relationship will be more complex. If the acceleration
a is uniform (i.e. does not vary over the time of the experiment), then the change in an
objects position is:
"x = v o t + 1 at 2
where vo is the initial velocity of the object at the beginning of the interval t and a is the
uniform acceleration that the object is subject to. Because the acceleration is constant,
the velocity, vf, of the object at time, t, satisfies:
v f = v o + at (4)
v 2 = v o + 2a"x
where Equation (5) is derived by squaring Equation (4) and substituting in Equation (2).
If the acceleration, a, is itself changing with time, the equations of motion are more
complex. You will study the motion of the cart down the inclined air-track for which
equations (3) and (5) will (probably) provide a reasonably good description.
The setup is depicted in Figure 1. The air track will be elevated at one end. Data will be
taken using the Science Workshop™ software and two photogates, interpreted with the
sensor “Photogates (2).”
In the first part of this lab, you will measure the duration of time, t, that it takes for the air
cart to travel a distance ∆x = x(t) - xo on the air track. The distance ∆x is the distance that
the cart travels between the two photogates. You will repeat this six times for six
different distances, ∆x.
If the initial velocity, vo, and the acceleration remain the same during each run, then you
should have the form of Equation (3). The acceleration will be constant if the angle of
inclination, θ, remains constant and any frictional forces do not depend on the location or
speed of the cart. Additionally, if you start the cart from rest at the same point on the air
track and the position of the first photogate remains fixed during each measurement, the
initial velocity will remain constant. Different distances ∆x can be obtained by varying
the location of the second photogate.
By inspection, you might expect that the acceleration, a, depends upon the angle of
inclination, θ, of the air track.
In the space below, draw a force-body diagram of the cart and apply Newton's 2nd
Law (which is in your text) to find the acceleration of the cart in the direction of
You should find that the acceleration is given by Equation (6):
a = gsin " (6)
Keeping θ fixed is easy: place a block of fixed height h under the single-foot end of
the air-track. The first photogate must be kept stationary, approximately 20 cm from
the high end of the track.
You should measure seven reasonably spaced data points covering the entire length of
the air track.
For each distance ∆x in Table 1, find the time t by averaging over six trials.
Determine the angle of inclination:
mcart = ____________
h = ____________
L = ____________
θ = tan-1(h/L) = ____________
Consult the drawing on the previous page to make sure that you measure the proper h
Measuring the Time of Travel for Various Distances:
[Photogates (2); Table: Time between Gates 1 & 2]
Trial xo x(t) t Δx = x(t)
(Initial Gate) (Final Gate) (Time) -x
1 Position) Position)
Enter your data, ∆x and t, into a graphing program. From the previous discussion, we
know that the relationship should be modeled by Equation (3), so we expect that your
plot of ∆x and t should appear to resemble a polynomial of second order, as seen in
y ( x ) = a0 + a1 x + a2 x 2 (7)
Here, y and x refer to the y-axis and the x-axis, not physical quantities, and a0, a1, a2
represent the adjustable coefficients of the polynomial. They are adjusted by your
curve–fitting software in order to make the fitting curve best approximate your data.
It is up to you to decide whether the resulting polynomial is a physically reasonable
(or “meaningful”) fit to your experimental data.
Perform the fit and record the “best-fit values” for the coefficients below.
a0 = __________
a1 = __________
a2 = __________
Attach a copy of your data and the fitted line.
A comparison of equations (3) and (7) tell us that a0 should be 0, a1 should be vo and a2
g sin !
should be 2 . You have already computed the theoretical value of a using Equation
(6). A theoretical value of vo can be computed by using Equation (3). Complete the table
Experimental Theoretical % Error
acceleration, a ( )
initial velocity, v o ( )
Does your fitted value of ao agree with what you expect? Comment.
Does your fitted value of a1 agree with what you expect? Comment.
Does your fitted value of a2 agree with what you expect? Comment.
We can also study kinematics by measuring the initial, vo, and final velocities, vf. When
you measured the time that the cart spent between the photogates, the software also
measured the time required for the cart to pass through each photogate, ∆t. If the length,
∆L, of what passed through the photogate is sufficiently small, the quantity ∆L / ∆t
provides a good approximation to the instantaneous velocity of the cart as it passes
through the photogate.
The choice of the length, ∆L, is a compromise between two conflicting needs. If ∆L is too
small, then the transit time, ∆t, is too small and the percentage error in determining v(t) is
unacceptably large. On the other hand, if ∆L is too large, the measured speed is not a
good approximation to the instantaneous speed. Be careful in determining your value of
Explain your method in the space below.
If you were careful when taking your data earlier, you do not need to make any additional
measurements. (Aside: reading the lab write-up ahead of time does save you time in lab).
The table below will guide you through the measurements to calculate the acceleration
using Equation 4.
∆L = __________
Δ Δ1 ( ) Δ2 ( ) v1 ( ) v2 ( ) a cal ( )
x t (Time t (Time (Velocity (Velocity c
Gate 1) in
Gate 2) atGate 1) atGate 2)
aaverage = _________
i. What is the magnitude of the force exerted on the air cart by the air from one of the
holes in the air track (i.e., the force lifting the cart up)?
ii. Is the assumption that the air track is frictionless valid? Support your answer using
the data you took in this lab.
Design an experiment to determine the extent of frictional effects on the air
track. Do this experiment and present your data and analysis.
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