Today we

• demonstrate the analysis of the acceleration vs. ramp slope experiment
• discuss the various sources and types of errors encountered in the experiment
• use a flow diagram to analyze the uniform acceleration situation when we are given v0, vf and `dt, symbolizeing each new quantity in terms of the original three quantities and obtaining in the process the two most fundamental equations describing uniformly accelerated motion
• use a flow diagram to analyze the situation in which we are given v0, `ds and `dt, without bothering to symbolize each result in terms of the original three quantities
• analyze with the use of the flow diagram the situation in which we are given `dt, a and v0, obtaining a third fundamental equation describing uniformly accelerated motion
• attempt to model the basic 'impossible situation', in which we are given v0, `ds and a
• devise a strategy for using the two most fundamental equations to reconcile the 'impossible situation'
• discuss the difference between the quantities v0, vf, `dv, vAve, `ds, `dt and a in terms of which we understand uniformly accelerated motion and the quantities v0, vf, `dt, `ds and a in terms of which we can symbolically analyze uniformly accelerated motion

We pose the following questions with regard to the experiment:

• How do flow diagrams help us see the structure of our reasoning processes?
• How do the two most fundamental equations of uniformly accelerated motion embody the definitions of average velocity and of acceleration?
• How can we interpret the third fundamental equation of uniformly accelerated motion?
• Why can we not directly reason out the basic 'impossible situation'?
• What strategy will we use to reconcile the basic 'impossible situation'?
• What is the difference between understanding uniformly accelerated motion and analyzing it with the use of equations?
• How do we extrapolate our acceleration vs. ramp slope data to obtain an estimate of the acceleration of gravity?
• How do the unavoidable timing errors due to the uncertainty in the computer timer affect our estimate of the acceleration of gravity?
• How could the slight slope of the table on which the ramp rests, if not accounted for, affect our graph of acceleration vs. ramp slope but not our estimate of the acceleration of gravity?
• How could anticipation of the instant at which a cart reaches the end of the ramp, but not of the instant at which it is released, affect our graph as well as our estimate of the acceleration of gravity?

With respect to the analysis of uniformly accelerated motion, the following questions arise:

• How do flow diagrams help us see the structure of our reasoning processes?
• How do the two most fundamental equations of uniformly accelerated motion embody the definitions of average velocity and of acceleration?
• How can we interpret the third fundamental equation of uniformly accelerated motion?
• Why can we not directly reason out the basic 'impossible situation'?
• What strategy will we use to reconcile the basic 'impossible situation'?
• What is the difference between understanding uniformly accelerated motion and analyzing it with the use of equations?

When we observe the acceleration of cart rolling down ramp vs. the slope of the ramp

• For reasons that we will see later we expect that for small slopes and for negligible friction, the graph will be approximately linear (this is what is expressed in the smallest circled comment on the graph; this comment is probably hard or impossible to read).
• The slope of this graph should be very close to the acceleration of an object falling freely under the influence of gravity, again for reasons we will detail later.
• It turns out that for small slopes, friction has a nearly constant influence on the acceleration, so that the assumption of negligible friction is not necessary for the above conclusions.

On the graph below

• We see three large dark blue dots representing the accelerations at three different ramp slopes.
• We have sketched a red line, more or less straight, which comes as close on the average is possible to the three data points.
• This straight line represents our best estimate of how the acceleration of the cart changes as the slope of the ramp changes.

We easily find a slope of this straight-line graph by picking two points on the line. Note that the straight line in this case doesn't go through any of the data points.

• There is no reason to expect that any of the data points are completely accurate, because of timing errors and other uncertainties in our observations.
• So there is no reason to try to make a graph go through any of the points.
• The line should come as close as possible to the data points, on the average, but need not pass through any point.

There are a number of unavoidable errors in this experiment.

• One common error is that the individual timing the motion of the cart down the ramp is looking at the cart while doing the timing.
• What typically happens in this situation is that when the car is started, the timer is started slightly later, due to the reaction time of the person doing the timing.
• This would be OK if the individual's reaction time caused the individual to stop the timer with the same delay. 
• However, the person doing the timing often anticipates the instant when the cart reaches the end of the ramp, so that the delay is not added onto the end time as it was to the starting time.
• In fact, the anticipating individual often triggers the timer slightly before the cart reaches the end, compounding the error even further.

This is a serious error, because the time interval `dt is used twice to calculate acceleration.

• We use `dt in calculating the average velocity from which we determine the final velocity and therefore the change in velocity
• We use `dt again in dividing the change in velocity by the time interval to get the acceleration.

We call this a 'systematic error', because it tends to be about the same on each timing and is therefore perpetrated throughout the data set.

Another possible source of error is the fact that the computer timer is only accurate to within about .03 seconds.

• This uncertainty affects the placement of points on the graph, and hence affects the slope of the straight line approximating the graph.
• This error, however, is random and not systematic -- it can result in a timing that is either too long or too short, with equal probability.

A third source of error could be the slope of the table on which the ramp rests.

• If we fail to take this slope into account when we measure the altitudes of the two ends of the ramp, relative to the table, the slopes we calculate from these measurements will all be incorrect by an amount equal to the slope of the table.

http://youtu.be/oafK7QGrtr8

The figure below shows the effects of three factors on the graph. The blue points, labeled 'ideal correct acceleration', represent the actual acceleration vs. slope, which we assume is somehow known to us.

One factor might be the slope of the table on which the incline rests.

• If instead of using a level we determine the slope by measuring how high each end of the ramp is above the table, and use the difference as the rise of the ramp, we might neglect to account for the fact that the table might not be level.
• It would seem that this would throw off our final result, which is the slope of the graph. However, the only effect this systematic error has on the graph is to change the m coordinate of each point by an amount equal to the slope of the table, which is always the same.
• This shifts each 'ideal' graph point horizontally by the same amount, but it is clear that this has no effect on the slope of the graph.
• Since it is the graph slope that comprises our final result, a small table slope would have no effect on our conclusions. 
• The points corresponding to such a shift are indicated in red.

Flow diagrams can help us to obtain formulas relating the basic kinematic quantities in terms of which we have been analyzing uniformly accelerated motion.  These quantities are v0, vf, `dv, vAve, `dt, `ds and a.

For example consider the situation where the velocity increases from 13 m/s to 33 m/s between clock times t = 8 s and t = 12 s.

• From this information we see that we know or can easily determine the time interval `dt, and initial and final velocities v0 and vf.
• These three quantities are represented in blue at the top of the diagram in the figure below.
• The other quantities obtained from the three given quantities, and quantities we can determine from the quantities obtained, will be shown at successively lower levels of the diagram, with lines indicating the logical 'flow' from the quantities used to the quantities obtained.
• In this example, each new quantity will be obtained from two quantities which have already be determined.  This is the ideal situation, since we can easily conceptualize each step in such a diagram.  However, in some cases three or more quantities will be required to obtain a new quantity.

As we often do, we ask the question 'what can be determined from what we know at this point?'.

• We see that from v0 and vf, we can determine `dv = vf - v0, and we indicate this in green at the next level of the diagram.
• From v0 and vf we also determine the average velocity vAve = (vf + v0) / 2, which we indicate in light blue.
• From the change in velocity `dv and the time interval `dt we determine the acceleration a = `dv / `dt.
• Since `dv = vf - v0, we see that a = (vf - v0) / `dt. Note that this is the definition of acceleration.
• This equation can be rearranged into the form vf = v0 + a `dt, which we write in black and circle.
• This is the usual form of one of the two most fundamental equations of uniformly accelerated motion.
• Note that this equation expresses the definition of acceleration.

http://youtu.be/Yl5wM7vqBeI

The figure below depicts a situation in which `dt, `ds and v0 are known.

• At the first level below the top, we indicate that the average velocity is the displacement divided by the time interval, vAve = `ds / `dt.
• We then combine the average velocity with the initial velocity, using the fact that for uniformly accelerated motion the initial, average and final velocities are equally spaced, and we obtain the final velocity vf.
• At the third level we use initial and final velocities to get the change in velocity `dv, which we finally combine with the time interval `dt to obtain the acceleration.

http://youtu.be/fPE4ftf8BnA

The figure below shows the basic 'impossible situation' in kinematics. It is impossible in the sense that we can't reason out the solution directly, as we did in the examples above.

• We are given the displacement `ds, the acceleration a, and initial velocity v0.
• We cannot draw any conclusions from any pair of these quantities without knowing something about the ones we don't know.
• Knowing displacement and acceleration, we cannot find `dt, vf, `dv or vAve, so knowing these two quantities gets us nowhere.
• Knowledge of acceleration and initial velocity isn't worth anything without knowledge of the time interval.
• Knowing displacement and initial velocity doesn't tell us anything either.
• So we seem to be stuck.

We can get unstuck if we write down the two most fundamental equations and see what we know.

• Looking at the equations, we see vf is common to both. 
• We can therfore eliminate vf by substituting the expression vf = v0 + a `dt into the second equation.
• This will give is an equation involving only `ds, v0, a and `dt.
• These quantities are indicated in red, and the known quantities `ds, a and v0 are underlined, leaving only `dt has an unknown quantity.
• We can thus solve for dt in terms of the known quantities.