Student: I have figured out how to read simple graphs, and I hope it will help me to learn how to make
new graphs from old.

Mentor: What do you mean?

Student: I have seen people looking at a graph of distance vs. time, for example, and sketching the
graph of velocity vs. time for the same movement.

Mentor: Let us try to do it for the graph we used
before :

Student: I can already see how to make velocity graph for the straight line parts, because they
describe movement with constant velocity. For the first two seconds, the person moved with
velocity of 1/2 meter per second; during the fifth second, with velocity of -4 meters per
second (it is negative because the distance from zero was decreasing); and for the last three
seconds, the person did not move at all, so the velocity was zero. I can sketch that:

Student: But what about the time between two and four seconds from the start? I see that the person
was moving quite fast at the beginning of that time, but slowed down more and more.

Mentor: You can use approximations. You can divide this time period into several "pieces" and pretend
that velocity did not change during each of them. This is one of the most common methods of
calculus. The more pieces, or
intervals, you have...

Student: The better I will approximate the real situation. Let me do half-second intervals. From 2 to
2.5 seconds, the person covered about 1.5 meters... Wait, I better make a table:

Time

Distance covered

Velocity

2 to 2.5

1.5

3

2.5 to 3

.5

1

3 to 3.5

.25

.5

3.5 to 4

.1

.2

Mentor: You can already make an approximate graph of velocity for the whole time.

Student draws:

Mentor: I would like to point something out. Do you think it is possible or not to change velocity
instantly, as our graph suggests?

Student: I do not think so. It takes time to change velocity. You can't go from .5 meters per second
to 3 meters per second in no time, as our graph suggests. In real life, you will have to
accelerate from one velocity to another, even if you do it very fast.

Mentor: I suggest we smooth the graph out a little to make it more realistic, like this:

Student: Now we did it! We made velocity graph out of distance graph!

Mentor: Let us now take this graph and build a graph of acceleration by it. Remember, acceleration is
change in velocity.

Student: So if velocity does not change, acceleration is zero. When velocity increases, acceleration
is positive, and when velocity decreases, acceleration is negative. In our velocity graph, in
the period from 2 to 4 seconds velocity decreases fast at first and then slower and slower. By
the end of this period, velocity stays almost the same. So acceleration will go from some
negative number to almost zero, for time from 2 to 4 seconds. The places where we smoothed the
velocity graph out mean very fast changes in velocity, that is, large numbers for
acceleration. My graph looks strange:

Mentor: This only happens because the original movement is strange. By the way, is it realistic that
acceleration changes instantly?

Student: Not really, so we may need to smooth out the acceleration graph. It is interesting that in
every place where the distance graph is a straight line, the acceleration is zero.

Mentor: It makes sense, though.

Student: Of course it does: if the distance vs. time graph is a straight line, then velocity is
constant. Constant velocity means zero acceleration.