M.2.1 Measurements of Postition, Velocity and
is used with Pasco probes to demonstrate the kinematics of one-dimensional
M.2.1.1 Measurements of Postition, Velocity
and Acceleration of Constant Linear Motion
Equipment: Computer, 2.2m Pasco
track, Cart, Motion Sensor and Metal Slugs.
The above equipment is used in conjunction with
Data Studio to measure and plot the position, velocity and acceleration
of a Pasco cart as a function of time. During lecture the instructor
can show quantitatively that at each instant the velocity and acceleration
are the slopes of the line tangent to the position vs. time and the
velocity vs. time curves respectively. Data Studio can also be used
to obtain the average velocity and acceleration of the cart.
In the following Data Studio experiment, the
Pasco track was propped up slightly on one end with adequate metal slugs
to compensate for friction. Then the cart was given a slight push to
achieve constant velocity. The motion sensor was used to measure and
graph the cart's position as a function of time. The graph shows that
the position changes linearly as a function of time. A linear fit to
the position vs. time curve gives the slope to be 0.45 m/s. This constant
value corresponds to the average and instantaneous velocities for this
experiment. Furthermore, it is within experimental error of the mean
of the velocity vs. time curve (0.46 m/s). Taking the slope of the velocity
vs. time curve, we find that it is zero and therefore have zero acceleration
-- also demonstrated experimentally.
M.2.1.2 Measurements of Postition, Velocity and Acceleration of
Using the motion sensor in conjunction with Data Studio
and a basketball, we can study the kinematics of the ball under the
influence of gravity. The ball is thrown upward above the sensor and
its position is plotted as a function of time under the influence of
gravity. From the graph below we see that the ball's y-position changes
quadratically as a function of time. Taking the tangential slope of
this curve at each point gives a plot of the ball's velocity as a function
of time. The ball's velocity is a maximum as the ball is first thrown
upward, zero at the ball's maximum height, and negative its maximum
value when the ball returns to its initial starting position. By using
Data Studio's fitting algorithm to fit the velocity vs. time plot, the
slope of the linear fit is found to be -9.48 m/s/s as shown.
The acceleration graph is made by taking the tangential slope of
the velocity vs. time curve at each point and plotting this value
as a function of time. Data Studio is then used to calculate the mean
of the acceleration vs. time curve where we find the value to be -9.2
m/s/s. The values obtained by the linear fit to the velocity vs. time
fit from above and the mean of the selected data points of the acceleration
vs. time graph are in general agreement and close to the accepted
value of gravity of 9.8 m/s/s.
All these concepts can be demonstrated quickly and efficiently
during a classroom lecture.
M.2.1.3 Instantaneous Velocity, Average Velocity,
and Acceleration using the Air Track
Below is a sample set of demonstrations with an air track for illustrating
these concepts using a clock (the "white clock") that measures
the elapsed time for a glider to travel one meter and another clock
(the "red clock") that measures the time for the 0.1m flag
of the glider to pass a sensor at the end of the one meter interval.
1. First explain exactly what the clocks are measuring to the students.
A transparency is available that can be projected during the demonstrations
to remind them what is measuring what. Show them that the white clock
measures the elapsed time for the glider to travel one meter by passing
your hand through the start-gate, counting off "one thousand one,
one thousand two, " for several seconds, and then passing
your hand through the stop-gate. Then show them that the red clock measures
the time for the glider flag to pass its sensor by blocking its gate
with the glider flag, counting off several seconds, and removing the
Then if the white clock reading is labeled T and the red clock reading
is labeled t:
average velocity = 1 meter/T
instantaneous velocity = 0.1 meter/t
At some point you may wish to discuss how the exact instantaneous velocity
is defined in terms of the calculus derivative by imagining the flag
length to become smaller and smaller.
2. To check that everyone understands what the clocks are measuring,
ask them, "If the track is level and I send a glider through the
gates, what will be the relationship between the readings of the two
clocks? (and then do the demo!) Answer: red clock reading = 1/10 white
3. Now use a block to tilt the track up. Ask, "The red clock should
now read (greater than, less than, the same as) 1/10 the white clock.
In other words, is the instantaneous velocity at the end of one meter
of acceleration (greater, less, the same) as the average velocity over
the one meter distance?"
4. Now use a larger mass glider. "The clock readings should be
(greater, less, the same as) before?"
5. "At what fraction of the one meter distance does the glider
attain an instantaneous velocity equal to the average velocity over
the one meter? In other words, where should the red sensor be placed
to get red = 1/10 white with the track tilted?" (Answer: 1/4 meter)
6. You can check the measured acceleration against the tilt of the
track. If the track length is L and it is tilted height h,
acceleration = a = g sin q
Then from the white clock, a = 2 meters/t2 a = 2 meters/T 2
and from the red clock, a = (0.1m/t)2/2m = 0.005 meters/t2