The next mindset that we’d like to discuss
is proportionality. To be able to think about
physical variables in terms of simple proportions.
So for example one thing we’ll discuss is
how far something is following if we drop it
in a given amount of time. And if I ask you
whether the distance that this thing has fallen
depending on the time or the time squared, one thing
that you might be able to find out using our unit
analysis is that as we saw depends on time squared.
But supposed you have that as a given and I ask
you a new question. So I say that first we drop it
and we allow it to fall for a certain amount of time.
But now we drop it and we allow it to fall for
a new amount of time. Originally, it falls by
some constant times, the time’s squared as we saw.
So it depends on with proportional to the time
that you’re allowed to fall squared. But now
I ask you a question and say what happens if I say
double that time, I let it fall for twice as long,
what happens to the new distance that it falls
because I allow it to fall for a longer period of time.
What we can do is write the new falling distance
in terms of the old falling distance because
instead of writing constant times, the new
falling time, I can write constant times twice the
original falling time because we’ve said that
the new falling time is twice the original time.
Putting in this factor of two and seeing that
this whole quantity is going to be squared,
we can just do the squaring, we can square the 2.
2 square is 4, so we can just factor out the 4
after doing this and then we get to the one tricky
part which is up the entire thing to the right of the 4
that you see here is a constant times the original
falling time squared. And that’s just your
original falling distance that we see in the first line.
So now we can see that the new distance that it fell
is in fact just four times the original distance
that it fell. So one way to just restate everything
that we’ve just discussed here in a quick and simple way
is if I allow something to fall for twice as long,
it will fall 4 times as far. Understanding proportionalities
like the distance falling depends on the times squared
means that this is something that you would know
very quickly just by thinking in your head
If I double the time, what do I do to the distance
if the distance depends on the time squared.
and so if I multiply the time by 2, the distance
multiply by 4. If I for example multiply the time by 3,
the distance would increase by 9. Because we will
take 3 square and find 9 using the same method.
In this overview, we’ve discussed a basic idea what
the course is going to look like as we go through
and then there’s some skills that you'll want to know
for the future. We reviewed scientific notations, which I
highly recommend you practice often. We reviewed unit
analysis and how to think about the units of variables
in order to come to conclusions very quickly and easily.
We also talked about how to think about variables
simply in terms of what they're proportional to.
Whether they’re proportional to a thing,
or that thing squared, or that thing cubed
and how to find new variables, new values
based on the original values of variables
very quickly and easily.