Today, we’ll learn about some fundamental concepts in physics.
See, one of the most important functions of physics is to describe how things move.
But in order to do that, we have to be able to describe where things are.
So, in this video we’ll be exploring position, displacement, and distance.
If we want tell someone where the cookie jar is, it’s not very helpful to say “Oh,
it’s four meters to the left” if they don’t know where we are and what we mean
by “left”.
So the first thing we need to do is agree on a reference frame.
To do that, we need to choose a reference point as well as coordinate axes.
We call that reference point the origin, since it’ll be the origin of our coordinate axes.
Alright, let’s go ahead and choose one now.
In a 1-dimensional space, it’s simple to define coordinate axes, because we need only
one.
And look, now we can talk about left and right.
We can also do this in 2D space.
Let’s add an extra axis, like so, and now we’ve got up and down as well.
And in 3D, we would need a third axis, like so, and now we’ve got front and back.
Now that we’ve got a reference frame, it does make sense to say “Oh, it’s four
meters to the left”.
And that brings us to the most obvious thing we can do with a reference frame: describing
the positions of things!
So, let’s take a look at an example.
Here is a 1-dimensional reference frame.
If I tell you that a particle is four meters to the left of the origin, you now know exactly
where it is: right there.
I want you to notice something though.
I had to give you two pieces of information in order to specify the position of the particle.
I had to say that it’s four meters away from the origin, and that it’s to the left
of the origin.
In other words, I had to tell you its distance as well as its direction relative to the origin.
Those two quantities, distance and direction, together specify the position.
Here’s a good time to make a brief side trip into mathematics.
A mathematical quantity that’s described by a magnitude and a direction is called a
vector.
Vectors are kind of like arrows: they have a certain size or length, and they point in
a certain direction.
If this concept isn’t familiar to you, then we suggest that you watch our lesson on “Introduction
to Vectors”, for more context to this lesson.
So what’s this got to do with the position of a particle?
Well, when specifying the position of something, we need to mention how far it is, AKA the
distance, which is a magnitude, and the direction it’s pointed in.
So since position consists of both the properties required to describe a vector, we can say
that position is a vector!
Now you’re probably wondering: couldn’t we choose to represent left with negative
numbers and right with positive numbers?
Of course we could.
Then the particle we mentioned before would be at position negative 4 meters.
In two dimensions, though, direction is not so easily specified.
For example, here’s a particle located 1 meter south and 2 meters east of the origin.
We can’t represent the direction by a single sign in this case.
But what we could do is declare that north and east are positive, and that south and
west are negative.
Then we could represent the position as a list of numbers: (2,-1).
In fact, this list of numbers is just the coordinates of a point on a plot.
Great!
So now we know how to talk about the position of things relative to the origin.
Now, what if an object moves?
That brings us to the concept of displacement.
Let’s take a look at an example here.
A particle, initially located at negative 3 meters, moves to a new position at positive
4 meters.
We can ask, what’s the final position of the particle relative to its initial position?
That’s easy: it’s 7 meters to the right of the position it started at.
It’s easy to see this, but we can calculate it too.
Simply take the final position and subtract it by the initial position to give us 7 in
this example.
So, that gives us the displacement of 7 meters to the right.
Now, one thing we need to remember about displacement is that it doesn’t matter how the particle
moves from one position to another.
We’re only concerned with where it started, where it ended up, and the relative positions
of the two.
Here’s another way to think about displacement.
It’s simply the final position of the particle when we move the origin to its initial position.
So if we move the origin 3 meters to the left, we find that the final position of the particle
is 7 meters to the right of the new origin.
Good!
Now, let’s go ahead and look at a 2D example.
Here’s a particle that’s 1 meter north and 3 meters west of the origin.
Its coordinates are (-3,1).
It moves to a new position: (5,-2).
What’s its final position relative to its initial position?
It seems complicated, but let’s break it down.
In the east-west direction, the particle starts at negative 3 meters and ends at positive
5 meters.
Again, like we mentioned, we can find the displacement of this by taking the final position
and subtracting it by the initial position.
5 minus (negative 3) equals 8: that’s a displacement of 8 meters east.
Good.
And if we bring back the original graph and now focus only on the north-south direction,
we can do the same thing to get the displacement in this direction.
In this case, the particle starts at positive 1 and ends at negative 2.
So, subtracting the final position by the initial position, we have negative 2 minus
1 which equals to negative 3 and thus, it’s been displaced 3 meters south.
Therefore, the total displacement is 3 meters south and 8 meters east.
Great!
So looking back to the original graph again, we could have derived the same idea by shifting
the origin to (-3, 1), the initial position of the particle, which would give us a final
position of (8, -3).
Now, hmm, is it a coincidence that this Coordinate looks so similar to the displacement?
Well, absolutely not!
Because really, the displacement of a particle is its final position after we move the origin
to its initial position.
So could we possibly just write (8, -3) as the displacement?
Well, turns out we actually can!
So, you might think that, in the original coordinate system, there’s nothing at the
position (8,-3).
And you’re right.
However, we just need to remember that this particular list of numbers is not the coordinates
of a particle.
Instead, it tells us how much the particle moved.
Now, do keep in mind, just like position, displacement is also a vector.
And how can we tell?
Well, it encodes a magnitude, the straight-line distance between the initial and final positions
of a particle, as well as the direction from one position to the other.
Awesome!
The last concept we’ll focus on in this lesson is distance.
Now, there are two slightly different ways to use the word “distance”.
We can speak of the straight-line distance directly between two points, or we can speak
of how far a particle has travelled.
Distance in the first sense is closely related to position and displacement.
Remember that those are vectors: they include both magnitude and direction.
When describing the magnitude of a displacement vector, we used the word distance to describe
the direct separation between the initial and final points of a particle in motion.
Distance in the sense of how far a particle has travelled is straightforward, too.
Just be careful to distinguish it from displacement.
Take, for example, a particle that travels in a circle.
It starts at one point and ends at the same point.
The distance travelled might be 3 meters, but the displacement is zero, because the
start and end points are the same.
Now you might remember that quantities with magnitude and direction are called vectors,
while quantities with magnitude alone are called scalars.
Now, position and displacement are clearly vectors.
What about distance?
Well, distance only has a magnitude.
That means that distance is a scalar!
Alright, so let’s go ahead and test our understanding by doing some examples!
So, a particle begins at the origin, moves 8 meters to the right, then moves 3 meters
to the left.
What is the displacement between its initial and final positions?
Well, the first thing to remember is that displacement is a vector.
That means it must specify a magnitude and a direction.
So we automatically know that (a) and (b) are not options for our answer.
So how do we find the displacement?
Well, first we need to determine the initial and final position of the particle.
That’s easy.
We know the particle started at the origin which is 0, moved 8 meters to the right and
backtracked 3 meters, ending up at 5 meters making that our final position.
Finally, all we need to do is subtract the final position by the initial position, which
is 5 minus 0, giving us a displacement of 5 meters to the right!
Therefore, ‘c’ is the correct answer here!
Great!
Now instead, what if we were asked to find the total distance that the particle travelled.
How would the answer change?
Well, the answer would be ‘b’ in this case, since now we’re talking about the
total distance travelled.
When we want to find the total distance we have to consider both the movement to the
right and the movement to the left in our calculation.
So here we would just do 8 plus 3, giving us a total distance of 11 m.
And like we mentioned before as well, distance does not consider direction which is why ‘d’
would not be the correct answer here.
Awesome!
Well, that’s the end of this lesson on position, displacement, and distance.
Hopefully this will help you to think a little more precisely about how to describe where
things are.
So make sure to keep practicing some more problems to get a good grip on these concepts,
and we hope to see you in the next lesson.