Chapter 7. Slopes and Velocity

Distance Versus Time

Signed Distance along a Road

Constant Speed and Velocity Example

Henry 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 ·[km/hour]· Thus he travels

Here the slope m has the role of speed or velocity. That is,

everywhere on the graph. Observe that 

First Varying Velocity Example

Problem:   Graph the distance d to the origin of a path versus time t for the following journey of Harry Snail.

1. At two o'clock in the afternoon, he is 50 km west of the origin, he travels further west at 100 km/hr. He drives at this speed for 1[1/2] hours.
2. At half past three in the afternoon, he stops for one-half hour.
3. He then drives eastward at 75 km/hr for the next two hours and then stops for another 2 hours.

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

Footnote: When in doubt leave out, is a rule to follow in solutions of problems. Or, when in doubt say so, to show what is certain and what is not. Your credibility is at stake. Indicating precisely where you are guessing in a solution, identifies a question to be answered later by yourself or your instructor. And in marking assignments or tests, I would be less severe with mistakes explicitly identified as guesses than I would be with guesses deceptively presented as sure knowledge. Caution: Not all instructors will have this opinion.

2. The first described portion of the trip starts at the point A = (2 hrs,50km). He reaches the point B = ([3½] 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)

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 Velocity

Two 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 £ 3 km?

Questions

1. What happens when two different graphs have the same slope over the same interval?
2. Suppose you see two cars, one following the other and matching its speed exactly.
   
   (a) What would happen to the distance between the two cars?
   (b) Would the two cars travel the same distance in any given period?
   (c) How far would the following car travel if both started and stopped at the same time as the followed car?
   (d) How would you make the distance between the two cars change?
   
   
3. How would the graphs of the motions of the two cars be related if the motion of one always matched the motion of the other, but with a thirty second delay? In the same length of time, how far would each travel? Hint: How is the graph of d = f(t) related to the graph of d = f(t-a)?
4. Two cars travel along a straight road with the first car traveling twice as fast as the second, but still within the legal speed limit. Suppose the distance traveled by the first car is d₁ = f(t). Find a formula for the distance traveled by the second car d₂. Hint: First consider the case where both cars start moving at the same time.

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