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Suppose a car accelerates. What do you understand that we mean when we say, “At exactly 2.25 seconds after starting, the car is going exactly 64.8 $\mathrm{\frac{km}{hr}}$”? Does it mean that the car’s speedometer is pointing at exactly 64.8? It might, but the car can be traveling at exactly 64.8 $\mathrm{\frac{km}{hr}}$ even with a broken or inaccurate speedometer. Does it mean that the car’s number of km traveled is 64.8 times the number of hours it traveled? It cannot. It accelerated up until that moment, and at that exact moment the car hasn’t moved!

There are a number of meanings that we can give to “… going exactly 64.8 $\mathrm{\frac{km}{hr}}$”. One is that, for a very small interval of time around 2.25 seconds, the car moved essentially at a constant rate of 64.8 $\mathrm{\frac{km}{hr}}$. For that brief period of time we are unable, by any means, to discern a change in its speed.

Figure 4.3.1 shows a photo taken of a truck with the shutter open for $\frac{1}{1000}$ sec. It appears that the truck is stationary. However, when we zoom in on the truck’s tail light (Figure 4.3.2), we see tiny streaks in the photo. The streaks were made by the truck having moved slightly during the $\frac{1}{1000}$ sec that the shutter was open.

camera's shutter was open. A moment in time is a small interval of time.

Let

For the $\frac{1}{1000}$ sec that the shutter was open, and assuming the truck’s speed was essentially constant during that time, we can say two things:

- for d
*t*≤ 0.001 sec, d*D*= s d*t*meters, so

- when d
*t*= 0.001 sec, d*D*= s∙0.001 meters.

- the change in the truck’s number of meters driven (d
*D*)

- is its speed in m/sec (
*s*)

- times the number of seconds (d
*t*) that the shutter was open.

Reflection 4.5.

a) Approximately how fast, in
$\mathrm{\frac{km}{hr}}$, was the car traveling?

b) Study the video at this link after you finish. Compare your reasoning in 1.a with the reasoning portrayed in the video.

c) What is your interpretation of the video’s moral, “all motion is blurry”? How are differentials of distance traveled and time traveled related to this moral?

b) Study the video at this link after you finish. Compare your reasoning in 1.a with the reasoning portrayed in the video.

c) What is your interpretation of the video’s moral, “all motion is blurry”? How are differentials of distance traveled and time traveled related to this moral?

2. Express the change in the car’s distance traveled in terms of differentials of numbers of km it traveled and numbers of hours it traveled.

3. Sketch a graph of the car’s distance traveled (in km) with respect to the time it traveled (in hours) during the time this photo was formed, given that the car had traveled 35.7 km up to the moment this photo was taken. How would the graph change had the car traveled 146.2 km up to the moment this photo was taken?