# 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 or

*current 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.

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.

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