Defining motion
We may define “motion” as the change in the position of an object with time. But what do we really mean by “change in position”?
The Statue of Liberty in New York does not change its position with time, our definition leads us to conclude that it is not in motion – yet it is. The Earth is rotating on its axis, it is also revolving around the sun, even our solar system is revolving around the center of our Milky Way galaxy, and the whole universe is expanding somehow.
So what do we mean by “change in position”?
The thing is, “change in position” doesn’t mean much unless specified to a certain reference in which the “change in position” takes place. In our case, although the Statue of Liberty is indeed moving, it is in the same position relative to its immediate surroundings.
That sounds like a good point until we ask ourselves, “What do we mean by immediate surroundings”? We might say it is the environment that is close enough to the statue and attached to Earth, so that the moon, the sun, and planets revolving around the Sun are not considered as “immediate surroundings”.
But this is also lacking because the surroundings may involve things that may also be moving. When viewed from a moving vehicle or ferry, for example, the Statue of Liberty does not appear fixed in position. And it sure does not appear fixed in position to someone riding a bike on Liberty Island, although they too qualify as “immediate surrounding”
So, again we need to get a little specific by what we mean by immediate surroundings.
“Immediate surroundings”?
We are looking for things in our immediate surroundings whose position from the statue is fixed. Such things might include other statues, monuments, geographical structures, or even buildings – if we can guarantee that they won’t be moving anytime soon!
Let’s say we eventually conclude that the Statue of Liberty is always at the same position when viewed from the Trump Tower in Manhattan. So, we say, the Statue of Liberty is not changing position with time relative to the Trump Tower – therefore it is not moving.
A few cracks in this picture
It seems like we have tightened the lid on position, but let’s think about it. What if you are out there observing the Statue of Liberty from the Trump Tower and it suddenly “winks” at you. Allow me to point out that you are stone-cold sober while making this absurd observation.
While you are sure that, this is by all accounts a “movement”, your definition of motion does not register this as movement; the Statue is still at the same position from the Trump Tower as it was before the observation.
So we go back to our definition of motion and patch it up. And we can keep doing this until we are happy with our final definition, or until our patience runs out and just settle for what we would have. But for the sake of keeping this article within a reasonable length, let’s stop here for now. If we continue to fuss with the concept of position, we may never get started. For now, let us agree to say that an object is in motion if its position and (or) orientation changes with time relative to a fixed reference point.
So what is this article about?
We wish to know what motion is. In order to study motion, we must be able to precisely and intimately describe it – much like the way a painter describes their work. This is the main objective of this article; we wish to understand the concept of motion, precisely.
This process involves handing out some rigidly defined terms to familiar physical quantities and some … err mathematics. Unfortunately (or fortunately, depending on you), mathematics is an indispensable tool in describing motion. Logical reasoning might help us understand motion qualitatively, but mathematics is our best tool to describe motion quantitatively. We need it more than it needs us.
Now that we have an objective in place, let us proceed.
Description of motion: What do we mean by “position”?
Before we start talking about speed, displacement, acceleration and velocity, and other terms related to motion, we first have to define the position of an object in space 1. With that in mind, let us revisit our statue of Liberty example and try to tie some loose ends on our concept of position.
Emergence of a reference point
We went far enough in our previous argument to establish that we need some sort of a reference point if we are to accurately establish the motion of an object in space. The thing is, the only reference we have around is other objects. And these other objects may be changing positions relative to our original object, infusing chaos to our positioning efforts.
Luckily, we can always find objects whose distances from the object in question don’t change – or at least, they don’t change much. In our statue of Liberty example, we took the Trump Tower in Manhattan as our object of reference. We concluded, temporarily, that the Statue of Liberty wasn’t in motion (both translation and orientational) relative to this building.
Now let’s push this argument even further by introducing numerical values. The statue of Liberty stands at a distance of 6.3 miles, at 10° west of south from the Trump Tower in Manhattan. Since the statue is stationary relative to the Trump Tower, then it is always at a distance of 6.3 miles, 10° west of south from the Trump Tower.
Problems with a single reference point
Here are some concerns with this picture of position
Personally, this concept of position that we have developed isn’t really convincing. This may just be me being paranoid but hear me out.
Our whole concept of the statue of Liberty being stationary rests on only one object: The Trump Tower.
And I don’t like it.
I don’t like the fact that our entire concept of position is predicated on the fact that the statue is 6.3 miles, 10° west of south from the Trump Tower – and only the Trump Tower.
What if some terrible plate tectonic movement rips through New York, breaks away Manhattan (together with Liberty Island) and takes them to the Gulf of Mexico. Let us assume further that this plate tectonic movement leaves much of the city’s structures intact and without spatial orientation, so that the relative distance and direction of the statue of Liberty from the Trump Tower remains unchanged.
Now, while everyone on Earth can swear that the statue of Liberty moved we would still hold that the Statue of Liberty hadn’t moved at all. Even though it is now sitting at the Gulf of Mexico. Simply because its position and orientation hasn’t changed relative to our reference point. The statue of Liberty is still at a distance of 6.3 miles, 10° west of south from the Trump Tower – which is the same as the “last time we checked”2.
Multiple reference points
Maybe we should consider other multiple fixed points from which we can express the position of the statue of Liberty relative to these fixed points. So we might throw in a couple of buildings and structures into the mix. We could say the Statue of Liberty is 5 miles from the Empire State Building at 15° west of the north, and 2 miles from One World Trade Center at 12° west of south, along with our familiar Trump Tower reference.
We could say, the Statue of Liberty is always located at these relative positions from these places and if any of these position changes, then a movement must have occurred.
Problems with multiple reference point
That sounds convincing enough until we consider the possibility of a plate tectonic movement carrying all our reference points together as a unit. Which brings us back to square one.
We could pick an object so far away from New York such that, no plate tectonic movement can carry them both. For example, we could express the position of the statue of Liberty relative to Mt Kilimanjaro in Tanzania or Mt Everest in Nepal or the Great Wall of China in China or all of the three at once. But we could always make up an argument to undermine this ambitious approach. So we won’t go down that road.
But where does this leave us with our objective to “precisely and intimately” define motion – if we can’t even “precisely” describe position?
You could say nowhere, but to be fair with ourselves, we know more about position now than we did a while ago. Besides, it is the nature of definitions not to be completely satisfactory.
But this whole concept of position feels like one that is fundamentally puzzling.
Emergence of Frames of Reference
Millard Beatty Jr once wrote, “It is impossible to define everything because one term is defined by another term and the other term is defined by another and so on”.
It now feels more compelling than ever to state a somewhat similar statement, “It is impossible to find the absolute position of everything”. Because we find the position of one object relative to another object, whose position may be found relative to another object and so on.
Eventually, all positions (and motions) are expressed relative to what we call a frame of reference.
The conundrum with our plate tectonic argument is that we couldn’t stick to a single frame of reference. We believed that the statue must have moved, not because we observed it moving, but because we assumed that other observers on planet Earth must have observed it moving, so it must have moved, even though, relative to our reference point, the statue hadn’t moved at all.
Motion and position is all about perspectives.
So, perhaps our concept of position wasn’t so bad after all. Let’s revisit it one last time in light of this “perspective” angle.
We said the Statue of Liberty is at a distance of 6.3 miles, 10° west of south from the Trump Tower. We concluded that, if it was always at this position from the Trump Tower, then it wasn’t moving. What we need to add here is the crucial point that it is “not moving” with respect to the Trump Tower.
But that is not all, in our quest to find the position, we have intuitively brought forth the following interesting features:
- A reference point. Our reference point is the Trump Tower. This helps us establish the position of an object relative to this point.
- A coordinate system. This helps us locate the point or object in question. Our coordinates for the statue of Liberty are “6.3 miles, 10° west of south” from our reference point. This method of presenting coordinates is mathematically equivalent to the Cartesian coordinates of say, (-1.2, -6.2).
These features, compose what we know as the “Frame of Reference”
What is a frame of reference?
A frame of reference is a standard relative to which motion and position can be measured, it consists of an abstract coordinate system and a set of physical reference point(s) that uniquely determine the position of objects in that frame.
Position is always given with respect to a frame of reference. There has to be a coordinate system and a reference point to describe properly the position of an object. Even then, we can only speak of position relative to the given reference frame.
For example, geographically, the position of an object on Earth (such as the Statue of Liberty) is described in terms of degrees of latitude measured north and south of the Equator and degrees of longitude measured east or west of the great circle passing through Greenwich, England and at the poles. Here the coordinates are the latitudes and longitudes and the reference points are the Equator and Greenwich, England.
Motion and Frames of Reference
Now that we have defined position, let us move on to motion.
Well, we said “an object is in motion if its position and (or) orientation changes with time relative to a fixed reference point”. Since we now regard position terms of “frame of reference” rather than “fixed reference point”, it makes sense for us to modify our definition of motion to suit this amendment.
We therefore say that an object is in motion if its position or orientation or both change with time relative to a frame of reference.
It is apparent that motion and position are closely related to other, no pun intended. And we cannot describe them with absolute values, we can only speak of them in relation to a frame of reference. So, our objective to “precisely and intimately” describe motion of objects depends on what we choose to mean by “precise” – and that is a good place as any to stop.
