To date you have probably spent a significant amount of time and effort understanding uniformly accelerated motion.  At this point some students have pretty much mastered the process of applying the definitions, while most students are still occasionally inconsistent.

The problems posed so far can be solved by applying three principles, including two definitions and one property of a linear graph.  These are the basic principles of uniformly accelerated motion.  They are basic in that any situation involving linearly accelerated motion can be completely analyzed using these principles.

The three basic principles:

Three statements define uniformly accelerated motion and are the basis for the analysis of motion:

1. The average velocity of an object during an interval is its average rate of change of position with respect to clock time during the interval.

2. The average acceleration of an object during an interval is its average rate of change of velocity with respect to clock time during the interval.

3. If acceleration is uniform on an interval then the v vs. t graph is linear over that interval, which implies that the average velocity on that interval occurs at the midpoint of the interval, so that the average velocity is the average of the initial and final velocities.

These three statements are the basis of the analysis of the uniformly accelerated motion of any object, given sufficient information about that motion.

• Every time you analyze the uniformly accelerated motion of an object, you should base your analysis on these principles.  You should remind yourself with every problem how every step of your analysis is 'connected to' these principles. 
• This isn't easy to do at first, but once you have learned to make these connections, you will be able to do so quickly, easily and naturally. 
• Your understanding of uniformly accelerated motion will be a standard of comparison for your understanding of other aspects of this course.
• (These statements are also the basis for the analysis of nonuniformly accelerated motion, which is done first by trapezoidal approximation.  In the university physics course this analysis is extended to the use of calculus.)

Quantities which must be recognized and distinguished

To understand uniformly accelerated motion it is necessary to identify and distinguish the following quantities, and to understand in terms of the basic principles and definitions how they are related.

• v0, the initial velocity of an object on an interval
• vf, the final velocity of an object on an interval
• `dv, the change in velocity of an object on an interval
• vAve, the average velocity of an object on an interval
• `dt, the duration in time of the interval
• `ds, the displacement of the object on the interval
• a, the acceleration of the object on the interval.

Slopes and areas of graphs

The meaning of the slope and area beneath a segment of a graph are determined by the meanings of the quantities represented by the vertical and horizontal coordinates of a point on the graph.

The rise between two points is the change in the quantity represented by the vertical coordinate.

The run between two points is the change in the quantity represented by the horizontal coordinate.

The slope between two points is rise / run, which represents the change in the quantity represented by the vertical coordinate, divided by the change in the quantity represented by the horizontal coordinate. The slope therefore represents an average rate of change of the 'vertical' quantity with respect to the 'horizontal' quantity.  This interpretation usually makes sense as a rate of change.

The area of the trapezoid defined by a linear segment of a graph is equal to the area of the rectangle defined by the midpoint of that segment. The width of this rectangle represents the change in the horizontal coordinate, and the altitude represents the average value of the quantity represented by the vertical coordinate, so the area of the rectangle represents the product of the average value of the 'vertical' quantity and the change in the value of the 'horizontal' quantity. This quantity might or might not 'make sense', meaning that it might or might not have a significant interpretation.

Interpretation of slopes and areas

Interpretation of slopes and areas should be based on the above reasoning, not memorized on a case-by-case basis.

Application of these principles results in the following, which should be understood not as isolated examples but in terms of the above principles:

If we apply this reasoning to a graph of velocity vs. clock time we find that

• the slope between two points represents the average rate of change of velocity with respect to clock time, which is the average acceleration corresponding to this interval
• the area of the trapezoid defined by two points is the displacement corresponding to uniformly accelerated motion on the interval.

If we apply this reasoning to a graph of position vs. clock time we find that

• the slope between two points represents the average rate of change of position with respect to clock time, which is the average velocity corresponding to this interval
• the area of the trapezoid defined by two points is the product of average position and time interval, which is generally meaningless.

Fallacies and common errors

There is a lot of similarity in the calculations used in different aspects of the analysis of motion. 

• The way to avoid errors is to think in terms of meanings. 
• When you do a calculation make sure it makes sense in terms of the basic principles and definitions. 
• Never write down and do a calculation without thinking about what it means and why the calculation is as it is.

Confusing (vf + v0) / 2, (vf - v0) / `dt, (vf - v0) / 2, (vf + v0) / `dt

• (vf + v0) / 2 and (vf - v0) / `dt are important quantities. 
• (vf - v0) / 2 and (vf + v0) / `dt do not correspond to any important definitions.  (vf + v0) / `dt is completely meaningless, and (vf - v0) / 2 is half the change in velocity.  The latter might in some cases be useful, but it is not an important quantity.
• It is easy to confuse these calculations, but if you think about what the quantities mean and think in terms of definitions and basic principles, you will avoid these errors.

Average velocity is not (vf - v0) / 2.  You find the average of two quantities by adding them and dividing by 2.  To average two quantities you don't subtract them.  Correct statement:  If acceleration is uniform then average velocity is (vf + v0) / 2.

Average rate of change of velocity with respect to clock time is not (vf - v0) / 2.  Correct statement: The average rate of change of velocity with respect to clock time is (change in velocity) / (change in clock time) = (vf - v0) / `dt.

There are two ways to find average velocity of a uniformly accelerating object over an interval:

• The average velocity is defined as (change in position) / (change in clock time), which involves subtraction of two positions.  It does not involve subtraction of two velocities.
• If acceleration is uniform, you can average the initial and final velocities, which involves addition rather than subtraction.

If you subtract two velocities, you are finding the change in velocity, which is related to the average rate of change of velocity with respect to clock time, but which is not related to average velocity.