# How are speed limits calculated

## 1. Current speed

In our previous calculations, we simply divided the distance covered by the time required to calculate the speed. Strictly speaking, we only have one average speed calculated.
Let us assume, for example, that a car driver needs three hours to drive from Salzburg to Vienna (that's almost 300 km). The average speed is then about
\ [v = \ frac {s} {t} = \ frac {300 \, \ rm km} {3 \, \ rm h} = 100 \, \ frac {\ rm km} {\ rm h} \ ,. \]
Of course, he won't drive 100 km / h all the time. In the cities there is a speed limit of 50 km / h, while he can drive up to 130 km / h on the motorway. There are also construction sites along the way where the speed is limited to 80 km / h; maybe he got caught in a traffic jam on the way or took a break so that the speed has meanwhile dropped to zero. The instantaneous orcurrent speed (as it is shown e.g. by the speedometer of the car) will therefore always deviate significantly up or down from the average speed.
To get a good approximation for the current speed, we could divide the travel time into small time intervals \ (\ Delta t \) and measure the distance traveled for each interval. For example, if each interval is one second long, then \ (\ Delta t = 1 \, \ rm s \). If we have then determined the distance covered as \ (\ Delta s = 35 \, \ rm m \) in a certain second interval, then the current speed in this second is approximately
\ [v = \ frac {\ Delta s} {\ Delta t} = \ frac {35 \, \ rm m} {1 \, \ rm s} = 35 \, \ frac {\ rm m} {\ rm s } \,. \]
To convert that to km / h, we multiply that by 3.6 and get \ (v = 3.6 \ cdot 35 \, \ frac {\ rm km} {\ rm h} = 126 \, \ frac {\ rm km} {\ rm h} \, \). We call the small time and distance intervals here \ (\ Delta t \) and \ (\ Delta s \), because we want to express that the time and the path change by these small pieces in the second in question. The Greek \ (\ Delta \) (pronounced "Delta") always indicates a change in a size.
However, this is still not really an instantaneous speed. Our driver could have driven 130 km / h at the beginning and braked to 120 km / h within a second. In order to get more precise values ​​for the current speed, we could reduce the time intervals to tenths or hundredths of a second and carry out the above calculation again. However, we always only get an average speed over the relevant tenths or hundredths of a second.
We only get an exact value for the current speed if we let the time interval \ (\ Delta t \) approach zero in the limit value. Of course, the distance covered in this interval also approaches zero, but the limit value
\ [v = \ lim _ {\ Delta t \ to 0} \ frac {\ Delta s} {\ Delta t} \]
will remain a finite value that we define as the instantaneous speed.
This limit is strongly reminiscent of the definition of the first derivative
\ [f '(x) = \ lim _ {\ Delta x \ to 0} \ frac {\ Delta f (x)} {\ Delta x} \ ,, \]
only that the path \ (s \) now takes over the role of the function \ (f \), while the time \ (t \) behaves like the variable \ (x \) of the function. You can actually define it as follows: We assume that the path \ (s \) is a function of time. Then we write \ (s (t) \) to express this, and the function \ (s (t) \) will increase with time \ (t \) in our car driver example; unless our driver turns around on the way. In the worst case, \ (s (t) \) will remain on a plateau for a while if it stops. The instantaneous speed is then itself a function of time, namely the derivative of the path function \ (s (t) \):
\ [v (t) = s' (t) = \ lim _ {\ Delta t \ to 0} \ frac {\ Delta s (t)} {\ Delta t} \ ,. \]
Of course, in this case we need the path function \ (s (t) \) to calculate \ (v (t) \). This type of calculation is particularly important in mechanics, a branch of physics. Normally the distance function \ (s (t) \) is given in meters and the speed \ (v (t) \) in m / s.
The path function \ (s (t) = 5t ^ 2 -6t + 2 \) in m (meters) with the time \ (t \) in s (seconds) is given. How great is the current speed at any given time?

We get this from the derivative
\ [v (t) = s' (t) = 10t -6 \ ,, \]
in the unit m / s (meters per second). By inserting different values ​​for \ (t \) we get the current speed in m / s at all desired times.