| Composition Setup |
|---|
| Excerpt |
|---|
A plot of position as a function of time is an often useful diagrammatic representation of kinematics problems. |
Graphical Representation of Constant Velocity
Slope and Velocity
The mathematical definition of velocity is equivalent to the formula for the slope of a position versus time graph. To see the utility of this correspondence, consider the following position versus time graphs:
The first graph gives the position of some object (called object A) as a function of time. Looking at the graph, we can see that for each second of time that elapses, the object changes its position by 2 meters. This is the same as saying that the slope of the left plot is 2 m / (1 s) or, more simply, 2 m/s. Object A, then, is moving with a speed of 2 m/s.
In contrast, object B is only changing its position by 1 meter every second. Thus, it is moving with a speed of 1 m/s.
Finally, object C is changing its position by 2 meters every second, and so it has a speed of 2 m/s. Note that objects A and C have the same speed. The graphs are different, however, because object C is moving in the negative direction.
| Note |
|---|
That doesn't necessarily mean objects A and C are moving in different actual directions. When looking at position as a function of time, we should always consider what coordinate system applies (does the positive x direction point east? west? north?...), and we haven't determined that for any of the graphs yet. |
Intercept and Initial Position
Assuming the motion is defined to begin at t = 0, then the initial position of the motion (the position at time t = 0) can be found by taking the y-intercept of the position versus time graph. Thus, in the examples above, objects A and C begin thier motions at position x = 0, while object B begins its motion at position x = 5 m.
Graphical Representation of Constant Acceleration
Parabolic Form
From the formulas given in the model One-Dimensional Motion with Constant Acceleration, it is clear that a plot of position vs. time will give a parabola.
If the acceleration is positive the parabola will open upwards. The position at t = ti will be xi , as shown in the graph below (time at the origin is ti ):
|
In this case the position is positive. The fact that the plot of position vs. time is increased means that the initial velocity, vi , is also positive.
If the acceleration and the initial position xi were the same, but the initial velocity was negative , then the graph of position vs. time would look like this:
|
The parabola has a minimum value at the time tmin
| Latex |
|---|
\begin{large} \[ {\rm t}_{\rm min} = {\rm t}_{\rm i}- \frac{{\rm v}_{\rm 1}}{\rm a} \] \end{large} |
| Note |
|---|
This information is intended to familiarize the reader with the shape of the curve and how it behaves. Obviously, if the object starts out at time t = ti its real motion will not be described by the portion of the curve for t < ti, and so an object moving with positive initial velocity and positive acceleration will not have such a "minimum" position - it will move in the same direction, with increasing speed, for all t > ti . |
Concavity and Acceleration
Consider the position vs. time graph shown above. If you were to lay a ruler along the curve of the graph at the origin, the ruler would have to be horizontal to follow the curve, indicating zero slope. Thus, the velocity is zero at the origin. As you follow the curve, however, the ruler would have to be held at a steeper and steeper angle (see the lines added in the graph below). The slope grows with time, indicating that the velocity is becoming more and more positive (the speed is increasing). This positive change in velocity indicates a positive acceleration. In calculus terminology, we would say that a graph which is "concave up" or has positive curvature indicates a positive acceleration.
"Acceleration" versus "Deceleration"
In everyday speech, we distinguish between "accelerating" (speeding up) and "decelerating" (slowing down). In physics, both situations are referred to as acceleration (which can be confusing). It is possible to give an exact definition of deceleration, however. Deceleration occurs when the velocity and the acceleration vectors have opposite directions. "Acceleration" in the everyday sense (speeding up) occurs when the acceleration vector and the velocity vector have the same direction. The two cases can be distinguished graphically.
...
Graphs that Represent Speeding Up
...
| |
positive acceleration | negative acceleration |
|---|
Both the graphs that show "acceleration" in the everyday sense (speeding up) have slopes that are steepening with time. The only difference is that one of the graphs has a steepening positive slope and the other has a steepening negative slope.
...
Graphs that Represent Slowing Down
...
| |
negative acceleration | positive acceleration |
|---|
Both graphs showing "deceleration" (slowing down) have slopes that are approaching zero as time evolves. (Again, one has a negative slope and one has a positive slope.)
| Warning |
|---|
It is a very common misconception that a negative acceleration always slows down the object it acts upon. This is not true. It is important to note that a graph which has a negative slope approaching zero (slowing down) implies a positive acceleration, and a graph which has a negative slope that is steepening (speeding up) implies a negative acceleration. It may help you to remember that the concavity of the graph specifies the direction of the acceleration. |
| Wiki Markup |
| {composition-setup}{composition-setup} {table:frame=void|rules=cols|border=1|cellpadding=8|cellspacing=0} {tr:valign=top} {td:width=325|bgcolor=#F2F2F2} {live-template:Left Column} {td} {td} Because of the similarity of the definitions of acceleration and velocity, it should be clear that acceleration can be thought of as the slope of a velocity versus time graph, just as velocity is the slope of a position versus time graph. It might not be clear, however, that we can also see the effects of acceleration in a position versus time graph. !acceleration^accel.gif! Consider the position vs. time graph shown above. If you were to lay a ruler along the curve of the graph at the origin, the ruler would have to be horizontal to follow the curve, indicating zero slope. Thus, the velocity is zero at the origin. As you follow the curve, however, the ruler would have to be held at a steeper and steeper angle (see the lines added in the graph below). The slope grows with time, indicating that the velocity is becoming more and more positive (the speed is increasing). This positive change in velocity indicates a positive acceleration. In calculus terminology, we would say that a graph which is "concave up" or has positive curvature indicates a positive acceleration. !acceleration^acceltangents.gif! h2. The Danger of Deceleration It is important to discuss for a moment one problem with the specialized vocabulary of physics. So far, we have discussed three different aspects of motion. Each one can be discussed in terms of a _vector_ concept (magnitude and direction) or in terms of a _scalar_ concept (magnitude only). For instance, we discussed displacement, a vector, and distance, a scalar. For motion in one direction, distance is the magnitude of displacement. We discussed velocity, a vector, and speed, a scalar. If we are discussing instantaneous velocity, then speed is the magnitude of velocity. Our last quantity, acceleration, can also be discussed in terms of a vector acceleration or simply the magnitude, but for acceleration we have no special term for the magnitude. The vector is called "the acceleration" and the magnitude is "the magnitude of the acceleration". This can result in confusion. This problem is exacerbated by the fact that in everyday language, we often discuss _distance_, _speed_ and _acceleration_. The everyday definitions of distance and speed are basically equivalent to their physics definitions, since we rarely discuss direction of travel in everyday speech and these quantities are scalars in physics (no direction). Unfortunately, in physics, we usually use the term "acceleration" to refer to a vector, while in everyday speech it denotes a magnitude. The difficulties do not end there. Everyday usage _does_ make one concession to the vector nature of motion. When we discuss acceleration in everyday speech, we usually specify whether the object is "accelerating" (speeding up) or "decelerating" (slowing down). Both terms imply a change in velocity, and so in physics we can call either case "accelerating". The physics way of explaining the difference is: || everyday term || physics equivalent || | acceleration | acceleration and velocity point in the same direction | | deceleration | acceleration points in the direction opposite the velocity | To understand the physics definition, imagine a child on a playground swing. If you want to help the child swing faster, you must push them in the same direction as they are currently moving (so the acceleration of your push is in the same direction as the child's velocity). If you want to help them slow down, you must push them in the direction opposite their current motion (so that the acceleration of the push points opposite to the velocity). The difference between acceleration and deceleration (in the everyday sense) can also be illustrated graphically: || positive acceleration \\ positive velocity \\ "accelerating" || negative acceleration \\ negative velocity \\ "accelerating" || negative acceleration \\ positive velocity \\ "decelerating" || positive acceleration \\ negative velocity \\ "decelerating" || | !acceleration^pos1.gif! | !acceleration^neg1.gif! | !acceleration^neg2.gif! | !acceleration^pos2.gif! | Both the graphs that show "acceleration" have slopes that are steepening with time. The only difference is that one of the graphs has a steepening _positive_ slope and the other has a steepening \_negative slope. Both graphs showing "deceleration" have slopes that are approaching zero as time evolves. (Again, one has a negative slope and one has a positive slope.) {warning}It is a very common misconception that a negative acceleration *always* slows down the object it acts upon. This is *not true*. It is important to note that a graph which has a negative slope approaching zero (slowing down) implies a positive acceleration, and a graph which has a negative slope that is steepening (speeding up) implies a negative acceleration. It may help you to remember that the concavity of the graph specifies the direction of the acceleration. {warning} h2. {toggle-cloak:id=ch} {color:blue}Check Your Understanding{color} {cloak:id=ch} {include:Accelerate, Decelerate} {cloak} {td} {tr} {table} {live-template:RELATE license} |