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PURPOSE

To study the concepts of speed, velocity, and acceleration through graphical analysis.

MATERIALS  human subjects, motion detector w/interface, masking tape

RELEVANT EQUATIONS

DISCUSSION

We continue the study of motion by focusing on the time variation of the velocity and its rate of change, the acceleration. In graphical terms, the acceleration can be calculated from a plot of velocity vs. time. Consider the velocity plots illustrated below in Figure 4-1.

In plot (a), the velocity of the body does not change with time. The rate of change of the velocity, the acceleration, of such a body is zero. This is illustrated by the slope of the line that represents v vs. t, or v(t). As before, the slope is defined as the rise (Δv) over the run (Δt), and since we have zero rise everywhere on the plot, the acceleration is zero everywhere.

In plot (b), the body is speeding up since the velocity gets larger and larger as time elapses, and the curve that represents v(t) is a straight line. Calculating the slope is indicated for a particular spot on the curve where the rise and the run have been drawn. You will notice that the slope would be the same no matter where along the line we choose to calculate it. The acceleration is therefore constant for this plot. Actually, it was also constant for plot (a), with a value of zero. Any time the v(t) vs. t plot is a straight line (linear), the acceleration is constant during that interval. As you will learn later, any body under the influence of a constant net force will experience a constant acceleration. This kind of plot is typical of motion like free fall.

Next, in plot (c), we see that the variation of v(t) vs. t is curved. This presents a problem if we want to find the slope --- where's the straight line? The answer is to draw the tangent line at a given point on the curve. This is a unique straight line that touches the curve at one and only one point. The slope of the tangent line is the acceleration at that point. In this context, the acceleration is referred to as instantaneous, because it's the value at that particular instant. You can qualitatively verify this fact by sketching in tangent lines near t = 0 and near the maximum value of t that is plotted. The slope of the tangent line, and therefore the acceleration, starts out small, but continues to increase as time elapses. This motion is one where the body is accelerating at a faster and faster rate.

Bear in mind that as a vector quantity, acceleration also has direction associated with it. In motion along a straight-line path, we distinguish the two possible directions with a plus or minus sign.

Many of the graphs you generate in this experiment represent velocity vs. time. Remember that the vertical coordinate of the graph is how fast you are moving at some instant and the algebraic sign of this coordinate measures the direction of motion. As you perform the experiment, keep the following points in mind.

• How fast you move is speed (the rate of change of distance with respect to time).

• Velocity is a special term that incorporates both speed and direction. For motion along a straight line, direction is indicated by the algebraic sign (+ or -).

• In this experiment, velocity is considered to be positive if you move away from the detector and negative if you move toward the detector.

• The faster you move away, the larger is the positive number representing your velocity. The faster you move toward the detector, the larger is the magnitude of the negative number for velocity. For instance, a velocity of -4 m/s is twice as fast as -2 m/s and both motions are toward the detector.

• Acceleration is the time rate of change in the velocity. As such, it is proportional to the change in velocity vector, . It is a positive vector if the change in velocity is a positive vector, and negative if the change in velocity is a negative vector.