A Second Train

A Second Train...
Suppose that there is a second train, different from the one we studied in Designing a Speedometer and that we know that, upon leaving the station, this new train's distance at time t is d(t)=t² (in kilometers away from the station).
For example, when t=1 minute, it is 1 kilometer from the station; when t=2 it is 4 kilometers away; when t=1.1, it is (1.1)² kilometers from the station, etc. [such acceleration (do you see it?) can't continue indefinitely; but this model is valid for values of t during the beginning of a trip].

Here, the information we are given about the motion of this second train is of a different character. In Designing a Speedometer we had a chart which gives a finite number of readings (each reading is a pair, like t=3 and kilometers=55). Here, we are essentially given what amounts to an infinite number of (odometer) readings. This is because we can plug in any value for "t" that we want, and we will have a corresponding value for "d", the distance from our train to the station at that time.

Now, try to discover what the speedometer of this train reads when t=3. The suggestion is to proceed in several steps in order to make this discovery. Namely:

First, find the average speed of this train over the time interval "starting when t=3 and ending when t=4" (which we call "the time interval [3,4]").
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Next, find the average speed of this train over the time interval [3, 3.5].

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Then, find it over the interval [3, 3.1].

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Now, find it over the interval [3, 3+f] where f is some unknown constant (think of it as a fraction; you just need to know how to take (3+f)² and then the rest is the same).
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Please notice that all of your calculations (answers) in (a) through (d) are average speeds, in contrast to the fact you are trying to discover an instantaneous speed (namely the one when t=3). So none of those answers are exactly what we are looking for. Nevertheless, they provide us clues about the behavior of the train's speed when one looks,time-wise, "closer and closer to t=3".

Based on (a) through (d), give your (carefully reasoned out) guess as to what the speedometer read at t=3. Explain your reasoning in a paragraph with full sentences.


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It is important to understand that if one asks for a given train (or any other object), the question: "what is the average speed?" then the question only makes sense if your question includes a specification of a time interval, that is, only if you ask what the average speed is over a time interval which you specify. It does not make sense to ask "what is the average speed" but it does make sense to ask "what was the average speed of the train between 2:00pm and 3:30pm?" (or "between t=3 and t=3.4" if you are using "t" readings instead of "am" and "pm" actual clock times, which is what we are doing here).

It is also important to understand that average speed is not the same as your speedometer's reading: your speedometer's reading is of course your "instantaneous" or "instant" speed. The key is that a collection of average speeds, taken over smaller and smaller intervals "surrounding" a time t (like t=3) will give you better and better approximations of the instant speed at that time t (e.g. at t=3).

Challenges: (i) What is this train's speedometer reading when t=4? (ii) Suppose a third train has distance d(t)=t²+1 from the station at time t. Find the speedometer readings when t=2 and t=3. (iii) A fourth train's distance from the station at time t is d(t)=3t²+t. What is the speedometer reading when t=4? Include all your calculations (neatly organized) and full sentence narration explaining your reasoning and justifying all of your conclusions!