### Video Transcript

Which of the following most
correctly describes the difference between fundamental and derived physical
quantities? (A) Derived quantities can have
more than one unit, but fundamental quantities can only have one unit. (B) Fundamental quantities can have
more than one unit, but derived quantities can only have one unit. (C) Fundamental quantities can be
defined in terms of derived quantities. (D) Derived quantities can be
defined in terms of fundamental quantities. (E) Fundamental quantities were
proposed before derived quantities were proposed.

In this question, we are being
asked about the difference between physical quantities that are fundamental
quantities and those that are derived quantities. Let’s recall that a fundamental
quantity is a quantity that cannot be separated into more fundamental or basic
parts. So, for example, length and time
are both examples of fundamental quantities. In contrast to this, any quantity
that can be separated into more fundamental parts and therefore is not a fundamental
quantity is known as a derived quantity.

The statement in option (A) claims
that derived quantities can have more than one unit, but fundamental quantities can
only have one unit. Now we’ve just said that time is an
example of a fundamental quantity. And let’s recall that some units we
can use to measure time include seconds, minutes, hours, days, and years. Since time is a fundamental
quantity and we’ve just seen that it can be measured in a variety of units, then it
can’t be true that fundamental quantities can only have one unit. We know then that the second half
of the statement in option (A) is definitely wrong. And so option (A) cannot be our
answer.

Now, let’s move on to the statement
in option (B), which says, “Fundamental quantities can have more than one unit, but
derived quantities can only have one unit.” We’ve already seen that fundamental
quantities can have more than one unit. So this first half of the statement
looks good. However, the second half of the
statement claims that derived quantities can only have one unit. Let’s recall that speed is an
example of a derived quantity. Some examples of units we can use
to measure speed include meters per second, kilometers per hour, and miles per
hour. This means that the statement that
derived quantities can only have one unit can’t be true, and so the statement in
option (B) cannot be correct. Now that we’ve eliminated these
first two answer options, let’s clear them off the board to make ourselves some
space.

Now, let’s consider the statements
in options (C) and (D). Option (C) says fundamental
quantities can be defined in terms of derived quantities, while option (D) says
derived quantities can be defined in terms of fundamental quantities. Now, fundamental quantities and
derived quantities can each be expressed in terms of each other. For example, the derived quantity
speed is the distance moved by an object per unit of time. So this derived quantity speed can
be expressed in terms of fundamental quantities as the fundamental quantity length
divided by the fundamental quantity time. This equation can be rearranged to
say that time is equal to length divided by speed.

So now we’ve got a fundamental
quantity, time, expressed in terms of a fundamental quantity, length, and a derived
quantity, speed. However, since speed itself is
obtained from the quantities length and time, then expressing the quantity time in
terms of length and speed is really just saying that time is equal to length divided
by length divided by time. Now on the right-hand side, length
divided by length over time is just the same as length multiplied by time over
length. We can then slightly rearrange the
way that we’ve written things on the right-hand side here. So the equation now reads time is
equal to length divided by length multiplied by time. Since length divided by length is
simply equal to one, then this term cancels out. And then we’re just left with the
somewhat circular and not particularly enlightening statement that time is equal to
time.

So we’ve seen through this example
equation here that derived quantities can indeed be defined in terms of fundamental
quantities. We’ve also seen an example in this
equation here of how it’s possible to express a fundamental quantity in terms of
other fundamental and derived quantities. However, since the derived
quantity, which in this case is speed on the right-hand side, can itself be defined
in terms of fundamental quantities, in this case length and time, then this equation
isn’t really defining the fundamental quantity time in terms of derived
quantities. It’s basically just a more
convoluted way of making the circular statement that time is equal to time.

So while the statement in option
(D) that derived quantities can be defined in terms of fundamental quantities is a
correct statement, it’s not really so true to say that fundamental quantities can be
defined in terms of derived quantities, which is the statement in option (C). Let’s eliminate option (C)
then. And at this stage it looks like
option (D) may well be our answer.

To be sure of this, though, we
should also check out the statement in option (E), which says that fundamental
quantities were proposed before derived quantities were proposed. Now a lot of these physical
quantities were proposed a really long time ago. For example, the idea of length,
time, and speed being measurable quantities is so old that it’s really not known
when the idea first appeared. This means that we don’t really
have any way of knowing whether or not it’s the case that fundamental quantities
were proposed before derived quantities. So then whether or not this
statement in option (E) happens to be true, which really isn’t something that we can
know for sure, it certainly isn’t the defining difference between fundamental and
derived physical quantities.

This means we can safely eliminate
option (E). This leaves us with our answer as
the statement in option (D). The most correct way to describe
the difference between fundamental and derived physical quantities is to say that
derived quantities can be defined in terms of fundamental quantities.