M1.1 How to measure speed and velocity

To start any type of measurements in Physics we have to define units.


unit: the measure of quantity that is defined to be exactly 1.0


The unit must have dimensions. For speed and velocity, dimensions can be derived from Equation M1.3. They are: dimension of displacement per dimension of time. In the SI system, the basic unit of displacement is a meter and of time – a second. Therefore the standard unit for speed is 1.0 m/s (meter per second). This is also a unit for velocity. Speed and velocity have the same dimensions. It is customary to write dimensions of any quantity using square brackets,

[v] = m/s.

To measure the speed of a moving object we must measure the distance ΔS traveled by it during a chosen time interval Δt. Then the speed v can be calculated from the equation

v= ΔS/Δt             (M1.4)


We used here ΔS as the distance and not to emphasize that speed is a scalar quantity. Equatoin M1.4 gives the average value of the speed during the time interval Δt.


The average speed: total distance traveled divided by the total time spent to cover this distance.


There are some tricky things connected with the calculation of the average speed. You know how to calculate the average height of the student in the class or the average cost of a lunch eaten by 20 students. These are arithmetical averages from n different values and they are defined as

sum of all n values / n.

If there are 5 students and their heights are (in meters): 1.80, 1.75, 1.65, 1.90, 1.70, then the average height in this group is

(1.80 + 1.75 + 1.65 + 1.90 + 1.70) / 5 = 1.76 m

Now consider a car traveling during time t1=600s with speed v1=10m/s and then during time t2=200s with speed v2=20 m/s. What was the average speed of this car during the total time it traveled?

Do not try to calculate an arithmetical average – this would be wrong. Follow the definition of the average speed – total distance traveled divide by time spend on this journey.

Distance S1 traveled during first 600 s

S1 = v1*t

and

S2 = v2*t

The average speed is, according to the definition given by Equation M1.4

( S1+S2 ) / ( t1 + t2 ) = ( v1*t + v2*t ) / ( t1 + t2 )

Substituting numerical values for speed and time we get the average speed

(10 m/s * 600 s + 20 m/s * 200s)/ 800 s = 12.5m/s


If you followed the example with the height of students you would get 15m/s, which is the wrong result.

Let’s consider this very misleading example. The distance S from point A to B on the road is 1000m. A motorcyclist rides this distance in one direction with speed v1=10m/s and back with speed v2=20m/s. What was his average speed?

There is a temptation to answer – 15m/s, as this is an arithmetic average from these two values of speed. To calculate the correct value we must find the total time spend to travel the total distance. From Equation M1.4 we have

Δt= ΔS/v

so the time spent on going from A to B is

t1=S/v1

and time spent on going from B to A

t2=S/v2

The average speed calculated from Equaton M1.4 is

v= (S+S)/(t1+t2)=(S+S)/( S/v1+S/v2)

Substituting numerical values to this equation we will get

v=(1000m+1000m)/(1000m/10m/s+1000m/20m/s)=2000m/150s=13.33m/s

Always be careful when calculating the average speed.

Later on we will show how to calculate the instantaneous speed, that is the speed at a given moment of time. The instantaneous speed as well as instantaneous velocity are important quantities in Physics calculations.
To emphasize that the symbol represents an average value of a given quantity a short line is drawn above this symbol. So, with this convention, the Equaton M1.4 can be written

             (M1.4a)


There are some other units of speed in everyday use. Each of them is built from units of displacement and time. These units are “fit” to human feeling of distance and time. A moving car or train or airplane is usually traveling for hours and the distance traveled is measured in kilometers. In such a case it is more convenient to express the speed in km/h (kilometers per hour). In the USA the mile (=1609 m) is used to measure such large distances and speed of cars trains etc. is expressed in miles/hour, abbreviated mph or, for consistency with SI notation, as mi/h.

It is easy to recalculate units and it is often done if one wants to compare the speed of different objects expressed in different units. As an example we recalculate miles/h into km/h and then into m/s.

We simply replace each mile by 1.609km, then each kilometer we replace by 1000m and finally the hour was replaced by 3600 seconds.

Let’s find what is the speed of very good sprinter running 100 meters. The world record (as of August 2004) is 9.98 seconds, so a time of 10.00 seconds can be considered as a very good result.

In different highly specialized areas of Physics other units of speed and velocity are used, mainly to avoid very large or very small numbers or due to many years of tradition.

In everyday life we usually are dealing with speed. In problems in Physics concerning moving objects where knowledge of velocity (speed and its direction) is crucial, all displacements are described in a common frame of reference and all velocities are given as vectors.


Frame of reference is a coordinate system in which positions of objects are defined.


We will mainly use the Cartesian coordinate system, but for describing some problems in Physics other coordinate systems are more suitable. The Cartesian coordinate system is described and explained in the next paragraph: displacement and velocity the advanced level.


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