I have a rather basic, maybe even dumb, question. I was wondering why speed is defined as it is:
$s = d/t$
Of course, what the equation means is not too difficult to understand. However, there are many ways that d and t could be related, for instance:
$s = d + t$
I am not sure who the first person to define speed was, but I was wondering how they made the decision to define speed as distance divided by time.
The definition of speed (please, let me call it velocity hereinafter) is not random at all.
It seems you understand that it must depend on the distance $d$ and the time $t$, so I'll skip to the next stage.
Evidently (for a constant $t$) velocity increases if $d$ does; and (for a constant space) $v$ decreases if $t$ rises. That constrains the ways we can define it. For example, your example of $d+t$ is authomatically discarded. You could say $d-t$, that satisfies the growing conditions.
Then we apply the reasoning in the limit case. For a 0 distance, velocity must be 0 independently of time (unless time is 0 too), that discards any sums. If the time to reach the space is infinite, the velocity must be 0. That's forcing $t$ to be a denominator.
So we deduce it's a fraction, but how can we sure there are not powers of those quantities? We impose the linearity of space. It doesn't make sense that the velocity is different if you pass from 50 to 60, or from 70 to 80 in the same time. If all points in space are equivalent, there cannot be distinctions like these, so using the numerator $\Delta d$ guarantees that all points in space are equivalent. If it were $\Delta d^2$ the result would be different from 70 to 80 and from 50 to 60, for example. That's againts the obvious principle that we can set the origin where we want (we must be able to measure from the point we choose, as we do everyday with a simple ruler, placing it where we want). The same reasoning applies to time.
So they must be a fraction, and there cannot be other powers than 1. The only possible difference is a constant factor
$s=k \frac{\Delta d}{\Delta t}$
And this is what speed (or velocity) is, after all. The constant is actually the unit factor. It depends on what units you are using. I hope this is useful to you.
The measure of distance over time is useful in physics.
Like many useful measures, it was given a name; in this case speed.
It is nothing but a name given to rate of change of distance with time. If you know the speed and any other quantity (distance or time), then you can find the third one.
P.S. You can add only dimensionally same quantities. So $s = d + t$ is wrong.
Imagine you have a car. I travel a mile in the car. But in what amount of time? If I travel a mile in an hour, that's a very slow car. But if I travel a mile in a minute, that's a decent car.
Let's say we have a decent car, and it traveled a mile in a minute. How far could we go over an hour? Well, there are 60 minutes in an hour, so we go 60 times the distance we went in the first minute - 60 miles in an hour.
What we basically just did is set up a proportion - 1 mile corresponded with 1 minute, so what distance corresponds with 60 minutes? We write this out mathematically as $$\frac{1\text{ mile}}{1\text{ minute}} = \frac{x\text{ miles}}{60\text{ minutes}}$$
(You solve this by "cross-multiplying" - 60 minutes * 1 mile = x miles * 1 minute, and then we'd divide both sides by a minute, so here, basically the units just cancel, and we get 60 * 1 miles = 60 miles.)
Now, imagine we said we wanted to measure how 'fast' the car is going, and we'll call that speed. It's obviously a relation between distance and time ($d$ and $t$). We've already seen above that distance is proportionate to time, that is, it's represented by division.
Let's look at this a different way. If we travel a larger distance in a smaller time, the speed is higher. If we travel a shorter distance in a longer time, the speed is lower.
When we think about a number divided by another number, when the number on top (the numerator) is bigger than the number on the bottom (the denominator) the result of the division (the quotient) comes out bigger, like in 8 / 2 = 4 vs. 6 / 2 = 3. When the denominator is bigger, the result comes out smaller, like in 6 / 2 = 3 vs. 6 / 3 = 2.
In other words, division satisfies the properties the representation of speed needs to have - when $d > t$, $d/t$ (the speed) is large. When $d < t$, the speed is smaller.
A final way to think about it. We talk about a car's speed in miles per hour, or kilometers per hour. Miles/kilometers are units of distance. Hours are units of time. So we have $d/t$ again.
In short, speed is the rate of change of distance over time, and the equation is derived from calculus.
Strictly speaking, s=d/t is not true in general. Speed is the absolute value of the velocity, which is defined as the rate of change of the displacement with respect to time. For the 1 dimensional case velocity is given by:
$$v=\frac{dd}{dt}$$
Taking things a step further, acceleration is the rate of change of velocity:
$$a=\frac{dv}{dt}$$
Now, if you have no acceleration, the velocity can be calculated by solving the integral:
$$v=\int{dt}=C_{1}$$
Here, $C_{1}=v$, keeping things simple. The displacement is then:
$$d=\int{vdt}=vt+C_{2}$$
Now, if d=0 at t=0, $C_{2}$ must also equal zero, so:
$$d=vt$$
Or, equivalently:
$$v=d/t$$
Speed is the absolute value of this, i.e: $s=|d/t|$
If acceleration isn't zero, speed is $s=|at+v_{0}|$ where $v_{0}$ is the initial velocity. In this case it becomes awkward to define it in terms of the distance traveled. Acceleration can change over time as well, leading to a more complex relationship.
ds are really confusing. Mind if I edit them to roman style?
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Commented
Jul 31, 2017 at 11:03
When you're developing a physical theory, you're free to define your quantities as you like. You won't get away with $s = d + t$ since dimensions of addends don't match, but you can still come up with a whole bunch of equations, e.g. $s = d × t$.
In the end, physical theories are useful insofar they can describe the real world and predict what happens. Speed (or velocity) defined as $s = d / t$ is very useful for this: objects having the same velocity share a lot of interesting properties, like having a constant distance between them, or going from start to finish in an equal amount of time. Speed defined as $s = d × t$ just doesn't predict anything useful (or very little), that's why nobody defines it like this.
