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Home < Volume 3 Why Slopes - A Calculus Intro Etc << Chapter 7 Slopes and Velocity
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Chapter 7. Slopes and VelocityDistance Versus TimeSigned Distance along a RoadSuppose in traveling along a road, position at time t is given say by d = f(t). The coordinate d gives a signed distance to the origin or point of reference. Assume positions on one side of this origin have a positive d-coordinate and the positions on the other side of this origin have a negative d coordinate. The absolute value or magnitude of d, that is |d|, gives the unsigned distance to the origin or point of reference. The coordinate d will just be called the distance or signed distance hereafter.
Constant Speed and Velocity ExampleHenry Snail walks along a path away from its origin for four hours at a rate of 6 kilometers per hour. He started at 1 p.m., eight kilometers from the origin. Problem: Graph his distance d to the origin versus the time t since he began. Solution. His speed s = 6 kilometers per hour = 6 路[km/hour]路 Thus he travels
Harry travels a distance Dd = s Dt in time Dt. This information allows us to form the following table,
Here the slope m has the role of speed or velocity. That
is,
everywhere on the graph. Observe that
First Varying Velocity ExampleProblem: Graph the distance d to the origin of a path versus time t for the following journey of Harry Snail.
Solution.
The trip has five segments. Comments on each segment or portion follow. 1. Before his trip begins, he could be stationary, that is, not moving. This possibility, a suspicion which cannot be confirmed, is represented by the horizontal dashed line. The slope of this speculative dashed line is
2. The first described portion of the trip starts at the point A = (2 hrs,50km). He reaches the point B = ([3陆] hr, 200km) after traveling at 100 kilometer per hour for one and a half hours. The slope of this portion of the trip m = 100[ km/hr] = 100 km per hour. 3. The second described portion of the trip lasts for one half hour. By remaining stopped (stationary) for [1/2] hour, his (t,d) coordinate changes from B = (3[1/2]hr,200km) to C = (4hr,200km). The slope or speed m here is again zero. 4. By traveling at 75 kilometers per hour back towards the origin for two hours, his position coordinates (t,d) change from C = (4hr,200km) to D = (6hr,50km). The slope
5. Finally, he does not move for 2 hours. This gives the last portion of the graph with d = 50km and slope m = 0. Changing Units (Digression)The measure or unit of slope, speed and velocity in the last example is given by the ratio [ km/min] or km per minute, or km per one sixtieth of an hour. But other units (or ratio of units) are possible. The question becomes: what units do you like for the expression of these quantities. It is possible to change units of time and distance.
Multiplying by the number 1 does not affect the value of an expression,
but the number 1 can be written in many ways. Some, not all, are helpful.
These different ways can sometimes help in changing the units used for
the expression of a quantity. A few slope or velocity based examples
follow. Note that [60min/1hr] = 1 implies the following
Remark. An alternate method is to substitute 1 min = [1/60] hr into an expression. For example
Motion with the Same VelocityTwo small problems follow, to further examine the situation where two graphs have the same slope everywhere. The slope in distance versus time graph is given by speed or velocity. Information for the first problem. Paul starts at 3 kilometers north of his home. From there, for two hours, he walks northward at 5 kilometers per hour. Then he stops for a one hour rest. Next for one hour, he travels southward at 6 kilometers per hour. First Problem: How far is Paul from his starting point? Information for the second problem. John starts at a distance d0 north of Paul's home. He matches Paul's speed and movements. From his starting point, like Paul, he walks for two hours northward at 5 kilometers per hour. Then he stops for a one hour rest. Next for one hour, he travels southward at 6 kilometers per hour. Second Problem: How far is John from his starting point? Solution to both problems. A graph of Paul's motion is easily drawn with the help of the following table:
In this graph
Paul thus travels 7 km-3 km = 4 km from his starting point. A graph of both Paul's and John's respective motions can be obtained from the next table.
The distance of John from his starting point is also 4 km. The scale has been omitted from the vertical axis in the following graph since d0 is unknown.
This graph shows the case d0 > 3 km. How would the graph change if d0 was 拢 3 km? Questions
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Home < Volume 3 Why Slopes - A Calculus Intro Etc << Chapter 7 Slopes and Velocity
[1] [2] [3] [4] [5] [6] [7] [8] [9][10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
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