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Many times predictions are made based on our previous experiences; for example, how long will a car trip from Dallas to Houston take? Since the distance between the two cities is about 239 miles, such a trip should take about 4½ hours under normal traffic conditions. If in the first hour of driving only 30 miles are covered, what will the total time of the trip be? All these predictions are made, most of the time, based on previous experiences. Most people can make these predictions even before taking a formal course that include basic classical mechanics. In fact, the predictions are based on a simple proportion. The proportion is based on the ratio between the distance traveled and the time taken. This ration is the foundation for defining speed and velocity. Thus, these ratios are defined with the purpose of facilitating predictions about motion. In physics, a definition is a peoples (scientist) made concept that turn out to be useful for understanding experimental results. |
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Definition of Average Speed,
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.The average speed is a scalar quantity. Only the magnitude (value) of the speed is necessary for completely defining this quantity.
The bar over the letter v is the symbol used for representing average. Any quantity with the bar on top will be an average quantity.
The previous relation is an average value because the time taken can be long. Consider a person walking on a city street for several minutes, does the person maintain a unique walking rate? Since the answer to this question in most cases is no, the definition is really about average. The average speed is the rate at which distance is traveled in the time taken.
Units
The units of speed in the MKS system are the
. There are other
units such as feet/second, miles/hour, or Kilometer/hour but the one used in
these notes are the meter/second.
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Definition of Average Velocity,
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.The average velocity is a vector quantity. Thus, both the magnitude and direction must be given in order to determine the average velocity. In this case, the vector nature of velocity comes from the displacement, which is also a vector. The notation given below much a one dimensional system. However, the notation given can be extended to multiple dimensions. To simplify the understanding of vector, graphically, they are represented by arrows.
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The quantities represented in the last expression are also shown in the diagram. Again, since the time taken or the change in time are finite (not very small), the resulting velocity is an average velocity. The average velocity is the rate at which the displacement change with the time taken.
Comparing average speed with the average velocity it can be said that most of the differences comes from the difference between distance traveled (speed) and displacement (velocity); for example, the distance traveled can only be positive while the displacement can be positive or negative. Consequently, if there is actual physical motion, the average speed will not be zero. For the same case, the average velocity can be zero when the motion include displacement in opposite direction that can result in zero displacement.
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A tourist walk through New York City following the blue path of the figure, starting at the corner of W 42nd St. and Park Ave. he walks uptown along Park Ave. After 2½ hours he is at the corner of Park Ave. and W 110th St. walking East towards Frederick Douglas Ave. (Central Park W). He makes it to that corner in ½ a hour. Then, he continue walking toward downtown along Central Park W (8th Ave.) arriving at the corner of W 42nd St and Central Park W (8th Ave.) in an additional 3¾ hours. Before returning to W 42nd St. and Park Ave, he decides to calculate the distance traveled, the displacement, the average speed, and his average velocity. After an additional hour, the tourist is back at the corner of W 42nd St. and Park Ave. The scale shown in the picture is 1.0 Km. Thus, the magenta piece correspond to 0.25 Km. |
Before the tourist return to W 42nd
St. and Park Ave. and
In the same case at the previous question, find the displacement
of the tourist
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Calculate t
For the same path as in the calculation of the average speed,
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Units
The units of average velocity are the same as the units of average speed.
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Instantaneous Speed and Velocity |
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Conceptually, the instantaneous speed is the reading of the speedometer. The reading of the speedometer changes continuously with the changes in speed. The value of the speed is only valid during the instant when the actual reading is made. The instantaneous velocity is the reading of the speedometer, for the magnitude, and the reading of a compass for the direction. Thus, the instantaneous speed is the magnitude of the instantaneous velocity. The applet below shows a car starting from rest traveling from the origin, O, to a final destination at F where the car is stopped. The car speedometer will constantly change reading because of the change in speed required for the motion. Can we guess the reading of the car speedometer when the car is crossing the light pole? The first draw of the applet shows a method based on our previous definitions. To obtain a prediction, times and displacements are measured. |
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The reading of a watch are taken at two different instants, ti and tf; at those times, the car displacement can be identified by two points, xi and xf. Thus, the prediction for the reading of the speedometer can be obtained from the ratio between the displacement and the time taken, .
It can be expected that because of the nature of the motion and the simplicity of the measurement that this prediction will be off the actual reading of the speedometer. Can this prediction be improved? |
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A direct form of improving this prediction is by shortening the time involved in the previous calculation . In this case the new prediction is calculated from:
Since the time involved between the readings is smaller, the moving conditions for the car will change less. Thus, it is expected that the second prediction will be a more accurate value of the actual reading of the speedometer. A even better prediction can be obtained if the time between readings is reduce even further; for example, reduce the time taken to the time need by the car to cross the light pole . In this case, it is difficult to measure the time taken by the car for crossing the light pole. However, assuming that these times can be measured accurately, the prediction of the reading of the speedometer obtained from these data will be very close to the actual reading of the speedometer,
In conclusion, the quality of the prediction improves by shortening the time. The ideal situation will occurs when the time taken is closed to zero. Notes that small time taken also implies small displacement. Thus, the prediction are obtained from the ratio between two very small numbers. The ratio between two small numbers can be a large or small number; for example, the result is a large number for a ratio like 0.03 m/0.001 s = 30 m/s. |
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In mathematical notation, the previous processes is reduced to the following expression:
Notice that the bar for average is not used in the notation anymore, in the
limit, the velocity goes from average to instantaneous. In this notation,
, which reads
as the limit when delta t goes to zero,
represents the process of considering shorter and shorter time taken (Dt
smaller and smaller) until the time taken can be neglected. For algebra based
classes, the previous expression is just a notation to represent the process
of shortening the time explained above.
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Graphical Meaning of Average and Instantaneous Velocity |
Motion can be represented by two dimensional graphs. In particular, the graph of the displacement (not distance) versus the time can be very helpful for understanding the concepts of average velocity and instantaneous velocity.
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The sequence of drawing on the left represents a possible motion of an object. The motion is represented in a graph of the displacement versus the time, Graph I . In the horizontal axis the time is represented while the vertical axis represents the displacement. The graph can not represent the distance traveled because distances traveled are always increasing with time and not decreasing as it is the case here between 2 seconds and 4 seconds. How can be calculated the average velocity of the object between two points, i and f ? See the second graph, . For those two points, the horizontal and vertical coordinates can be read at the axis. Thus, the coordinates for the point labeled i are (ti , xi), Graph III of the sequence . And the coordinates of the point labeled f are (tf , xf), Graph IV of the sequence . The average velocity between the two points is calculated with the formula,
The previous expression is also the slope of the line between the two points labeled i and f in Graph V of the sequence . The next graphs of the sequence show the steps necessary to calculate the slope. The slope of a line is the rise over the run. In this case, the rise is shown in Graph VI . The run of the line is presented in Graph VII . Finally, the slope is presented in the Graph VIII . Therefore, the average velocity between two instants of time is the slope of the line between the two points of the curve representing the object displacement in a graph of the displacement versus the time. |
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Since the instantaneous velocity is obtained when the change in time gets smaller and smaller; graphically, this is represented when the points on the graph are getting closer and closer to each other. Thus, the average velocity calculated as the slope of the line between two points becomes the slope of the line for two points very closed to each other. The extreme case, Dt → 0, corresponds to the case when the line is tangent to the curve at the particular point. Therefore, the instantaneous velocity of a mobile at any time is equal with the slope of the line tangent to the curve at the corresponding time. The calculation of the slope for a line tangent to a point can be simulate using the applet above. This applet represents the first few seconds for the graph of the motion of a mobile. As always, the horizontal axis is the time, while the vertical axis is the displacement. The position of the vertical axis can be change by dragging the axis from the arrow head at the top. Similarly, the location of the horizontal axis can be changed by dragging the axis from the right most arrow head of the axis. Different points of the graph can be selected by clicking on the graph at the selected location. This applet can be used for calculating the slope of a line between two points in the graph. You can do so by clicking at two points of the graph and then clicking on the v Average bottom of the applet. If you check the Instantaneous velocity check box of the applet, a third point on the graph can be used for calculating the instantaneous velocity at the middle point, among the three points in the graph, when pressing the Run bottom of the applet. The simulation represent the process from where the instantaneous velocity can be calculated graphically. The results obtained do not pretend to be accurate but rather an illustration of the conceptual approach used for defining the instantaneous velocity at a given point. A more accurate result for the instantaneous velocity can be obtained by manually moving the line after the simulation is concluded. This is achieved by placing a point over the final line and dragging it to best fit the tangent to the second point. |