Position, Velocity and Accleration in a FanCart
Regina Bochicchio, Michelle Obama, Hillary Clinton
September 11, 2013
Purpose
The first purpose of this lab was to prove or disprove
the hypothesis that, for motion with increasing speed
in a positive direction with constant acceleration, the
magnitude and sign of the acceleration is identical to
that of the slope of the velocity time graph for the
motion.
A second purpose was to investigate the best-fit curve
for the position time graph made by the fan-cart and
find out whether the coefficients of the corresponding
equation represent any part of the actual motion.
Introduction
If an object moves with constant acceleration, it’s
velocity will change by the same amount every second.
The resulting velocity time graph is linear. The slope
of the velocity time graph can be obtained by using the
“rise over run” formula. Since velocity is on the
vertical axis and time on the horizontal axis, this value
can be calculated using the formula Δv/Δt, where the
bold indicates this is a vector.
The acceleration vector is defined as the rate of change of
velocity (Cutnell, 2013). This value can be calculated
by Δv/Δt. If this is true, then the average magnitude
and sign of an object’s acceleration, should be equal to
the to the slope of the velocity time graph.
Equipment
EQUIPMENT: Motion detector, USB cable, computer,
Logger Pro software, cart, track, fan attachment with 4
AA batteries, additional mounting bracket and elastic
band (not shown below).
The lab setup is shown below:
motion
detector
Fig. 1: The track, cart with fan attachment and motion detector was set up as shown for
this experiment.
Procedure
• Set up the cart, ramp and motion detector as indicated in
Figure 1. Make sure the ramp is level, and the fan is
securely attached.
• Turn on computer and start up Logger Pro. Open
Logger Pro Experiments folder to Additional Physics,
RealTime Physics, Mechanics, L02A1-1.
• The fan cart should be no more than 20 cm in front of
the motion detector at the start. The fan should be
directed towards the motion detector so that the cart
moves away from it on the track (see Figure 1).
Procedure (cont’d)
• Turn on the fan, release the cart and collect the data.
When you obtain a good data run, use Store Latest Run
(from the Experiment menu).
• Choose a portion of the acceleration-time graph (at least
10 samples) that is relatively horizontal. Select this
portion and use the statistics feature (STAT) to find the
mean acceleration.
• Select the same portion of the velocity-time graph and
use the linear fit function (R=) to find the slope and yintercept of the graph.
• Select the same portion of the position-time graph and
use the curve-fit function to verify that a quadratic
curve has a close fit to the actual data.
Velocity-time (top) and acceleration-time (bottom) graphs
Fig. 2: Velocity and acceleration graphs for fan cart speeding up in positive direction.
This graph shows the slope of the velocity graph taken from
the linear-fit function. The value is 0.2919 m/s/s. Mean
value for the corresponding section of the acceleration graph
is 0.2938 m/s2 .
Error Analysis for Purpose 1
|0.2938 m/s2 - 0.2919 m/s2 |
_________________________
0.2938 m/s2
= 0.006%
These numbers are very close in value!
Position as a function of time
Fig. 3: Position-time graph for fan cart speeding up in positive direction.
The best-fit curve for the position-time graph showing
increasing speed in the positive direction is quadratic. The
error term generated by the software (Root Mean Square Error)
is .001, showing a very good correlation with the actual graph.
The equation of this best-fit curve is:
x = 0.1499t2 + 0.3273t + 0.3241
Coefficients of position-time equation
Fig. 4: Position and velocity graphs for fan cart speeding up in positive direction.
A comparison of the “B” term for the position-time graphs
shows it to be very close to the y-intercept value of the
velocity graph for the same motion: 0.3272 vs. 0.3318 m/s.
This latter value indicates v0 of the fancart.
Conclusions
• The hypothesis given in the purpose is proven to be
true. For an object moving at a constant acceleration,
in a positive direction and increasing in speed, the
slope of the velocity-time graph is equal to the average
acceleration of the object.
• The position-time graph for this motion can be
represented by a quadratic equation of the form:
x = At2 + Bt + C, where B represents v0, the initial
velocity of the object.
• This lab can be improved by investigating other
motions (e.g. slowing down, moving in a negative
direction) to verify the connection between the
coefficient B and the initial velocity of the object.
Works Cited
Cutnell, John D. and Kenneth W. Johnson. Physics.
New York: John Wiley & Sons, 2013 ed.
Setup for formal lab report 2