When I started learning instantaneous velocity it didn't make sense to me. I don't understand in real life why we can't measure instantaneous velocity and therefore why we use this concept.
Or is this a mathematical concept and can it explain a falling object? How should I think about this to make sense?
You are absolutely right to be questioning this. When the calculus was introduced there were enormous debates about this very thing. If the average velocity over a period from $t$ to $t+\delta$ is $$\frac{f(t+\delta)-f(t)}\delta$$ then the instantaneous velocity at time $t$ is given by $\delta=0$. It is therefore $\frac00$, which is nonsense.
In general there is no such thing as an instantaneous rate of change.
To give a mathematical example, suppose that the position at time $t$ is $|t|$, the absolute value of $t$. Then the average velocity over any interval ending at $t=0$ is $-1$, the average velocity over any interval starting at $t=0$ is $+1$, and the average velocity over any interval centred on $t=0$ is $0$. Thus “the instantaneous velocity” at $t=0$ is $-1$ and it is $+1$ and it is $0$. Which is nonsense.
Learning when a function is “well-behaved” enough to make it safe to proceed to the limit $\delta=0$ took up a lot of the late 17th century. Because we have learnt the rules, we have a natural instinct for functions that are “well-behaved”, “safe”, or whatever, and as long as we stay within that playpen, we can pretend that instantaneous velocities exist and not come to any harm. But you are absolutely right: there is a whole stack of assumptions about the physical world, and the mathematical functions that describe it, that are necessary to make instantaneous velocities make sense.
Fortunately, in physics, these assumptions hold. On the one hand, if you are driving down the road at $30$ mph ($50$ km/h) and hit a lamp-post, you could say that at that instant your speed has dropped from $30$ mph ($50$ km/h) to $0$. On the other hand, in physics nothing happens instantaneously. You do not collide with the lamp-post all at once. First the bumper hits, and deforms, and slows you down a bit. Then the engine hits, and deforms and compresses, and slows you down a lot more. On a timescale of milliseconds, the crash is a gradual process and not instantaneous at all - so that the question “what is the velocity?” does always, millisecond by millisecond, have an answer.
So to summarise. The concept of instantaneous velocity can only make sense if the function of “position, given time” is sufficiently well-behaved - mathematically, if it is differentiable. Differentiability is not something that is automatically given to you for free: it is a cheerful fact about the physical world.
So now, knowing that instantaneous velocities don’t make sense in themselves, but do make sense because of how the world is, you should be able to carry on doing physics as if they really existed.
And do carry on questioning the fundamentals. One can get a long way by just believing what one’s teachers say just because they say it, but ultimately that means nothing but being an educated parrot. It is much better to dig down, and believe things because they are actually true.
Velocity is defined as distance travelled per duration time. Thus in order to measure a velocity you hand to observe for some time $\Delta t$ and you only get an average velocity for that duration and that distance. The instantaneous velocity is the velocity you get when you make the duration measured infinitely short:
Let $d$ be a place, $\Delta d$ a small distance and similarly $t$ be a time and $\Delta t$ a short duration, then:
$$v = \frac{\Delta d} {\Delta t} $$ $$ v(t) = \lim_{\Delta t\rightarrow 0}\frac{(d + \Delta d) - d} {(t + \Delta t) - t} $$
Here's a way to look at it.
Average velocity corresponds to the total displacement vector scaled by (resized by, multiplied by) $1/\Delta t$.
It corresponds, in direction and magnitude, to the displacement vector that would result from the object moving at that average velocity for $1s$. (This very arrangement allows you to scale it for arbitrary time values - this is why multiplying by time works to give you total displacement for that amount of time, assuming the average velocity is constant.)
Instantaneous velocity corresponds, in direction and magnitude, to the displacement vector that would result if the object traveled for $1s$ without changing its current speed and direction. It is tangent to the path - it points straight ahead of the object at every moment. So, at every point, it tells you which direction you're going towards, and how fast.
Now, as you shrink $\Delta t$, the average velocity gets closer and closer to the instantaneous velocity. Mathematically, there's a way to figure out what this vector tends towards (but never quite gets to be equal to) as $\Delta t$ approaches zero (but never quite gets there). So, instantaneous velocity is the limiting vector that the average velocity approaches to as $\Delta t$ approaches zero.
One way to think about it is that every point on the path is "labeled" with an object that encodes current speed and direction (this object being the instantaneous velocity vector). Or, maybe think of a car or a plane with a speedometer and a compass (a heading indicator).
In a reverse mathematical process (integration) you can figure out displacement if you know instantaneous velocity at every point; you basically take small chunks of time, and multiply each with the instantaneous velocity at the start of that chunk (and you assume it doesn't change for that small duration), to get a small displacement vector for each chunk. You then add all those vectors up. And then, in another limiting process, as those time chunks get smaller and smaller, the resulting total displacement approaches the true value, and there's a way to figure out what that true value is.
Velocity is the rate of change of the position of an object.
Change, by its everyday as well as physical and even mathematical meaning, implies that you need to compare two things to each other - in this case it is, for an object in motion, first the position of the object at some given time, and then at a given later time. Hence, the difference you can measure between those two positions at different times are your "change" (expressed as a number in meters, kilometers, miles etc.).
Dividing the change (the distance driven by your car) by the time it took to do so gives you a number which we define as velocity (expressed as kilometers per hour, or miles per hour, or meter per seconds, etc.).
So far so good. This is the non-instantaneous kind of velocity which should be no problem and is not what you are asking about.
The instantaneous velocity is an idealized, mathematical concept; not something that occurs in reality. It would be like making a single photograph of a car and asking how fast the car goes. It is not possible to say - there is not enough information.
But what you can do is go back to your original non-instantaneous velocity and make the time between your two measurements ever smaller. You might start out with an hour, go to a minute, then to a second, then 1/10 of a second, and so on. For each of these ever smaller time spans, given that your measurement apparatus is accurate enough, you can find the two positions of the car. All of these different measurements give you possibly different velocities - i.e., the car may be driving slow during the first half hour, and fast during the second. If you measure over the whole hour, you will end up with some average speed. But if you measure only during a minute near the end of the drive, you might end up with the faster speed. All of this is totally normal and what we do all day, every day, when judging how long it will take us from A to B when commuting.
Now we come to the trick: let's make the time ever smaller, and keep either the first or second of the two timestamps fixed. Since cars do not accelerate that quickly in normal circumstances, you will end up with always nearly the same velocity, down to measurement errors. E.g., if you measure your non-instantaneous velocity over a time of 1s or of 0.5s (with the same start time), the car will have driven twice as far in the first measurement, but dividing the displacement by the time still gives the same result since the first timespan is also twice as large.
If you take this to the extreme, you end up with a (still and always non-instanaeous) real, physical velocity which is not changing at all between further finer measurements with one of the timestamps fixed, within your measurement accuracy. At this point, you define this to be the instantaneous velocity at that point in time (the point which you held fixed).
If you are a normal person, you're done and can go on with your happy life. If you're a mathematician, you spend the next half of your life figuring out how to create a formal mathematical mechanism to work rigorously with all of that, and eventually call it "calculus", much to the chagrin of generations of young students wrestling with those ideas!
Here is one way to look at it: Suppose we have a displacement time graph, say, of a race car, this graph is definitely not linear, at different points in time the engine is applying different forces on the wheels so sometimes the car is going fast or sometimes slow, let’s say you freeze time at a particular time $p$ and remove all forces acting on the car, by Newtons law $F=ma$, the car will travel at a constant velocity and that velocity is the instantaneous velocity at time $t=p$.
I motivate [instantaneous-]velocity as "the slope of the tangent line to a postion-vs-time graph", which is of course "the time-derivative of the position function". Then, make contact with the typical interpretations of velocity.
While there is some mathematical analysis that one could do (e.g. limit of the slope of chords or secant lines) to arrive at that, I prefer the slope interpretation.
I dislike the typical first introduction of "velocity" as "displacement over time",
because that is really so-called "average velocity"---really "time-weighted average of instantaneous velocities"... but I haven't defined them yet... since that's what I am trying to do [via this approach].
