Creation of the term Speed

When there is a translational motion, the object moves some amount of distance in some amount of time. The motion can be described by how fast its position changes. The change depends on the size of the path namely distance covered in the motion and the time taken for the object for this distance.    

Let’s say the relation between this distance and time is Speed. So, speed is the function of distance and time.

But can it be related with some mathematical operations?     

Let’s say speed as S and

            the distance covered in the motion as D and 
            the time taken for this distance as T.

It can be defined for unit of time as follows,

            To know what happen to the motion when one of these factors of variables change in the function of Speed, we should make one variable as a constant and change the other variable.

Making the time constant, we changed the distance. For a fixed time if the distance is too high it means the object moves very fast. If the distance is low, then it implies the object moves at a slow rate. Thus we found that the function Speed is directly proportional to the distance covered by the object in some amount of time.

Similarly, if you make the distance constant and changed the time taken for this distance, we can know that,

            If the time taken for some distance is high then it was understood that the motion happens at a slow rate and if the time taken for that distance is low then the motion happens quickly.

            Thus speed is indirectly proportional to the time taken for the distance in any motion.

            Hence, the Speed is directly proportional to the distance covered and indirectly proportional to the time taken and it can be simply written as

Speed  α  Distance and

Speed α   1/Time  [where “α” – “alpha” means “proportional to”].

Combining the above two, we can rewrite it as,

Speed α  Distance / Time

and let us take the proportionality constant as 1.

Finally we described the first mathematical term in motion as

Speed (s)= Distance (d) / Time (t) or s= d/t

We can get the meaning of this term d/t as follows,

“d” distance was covered in time “t”.

            consequently,

time “t”                        for        distance “d”               

[Note that d, t are the number values of distance and time]

            value of t*unit time      for        distance “d”

            unit time                       for        distance equal to the (value of d)/(value of t)

            unit time                       for        (d/t) distance

Then the length covered in unit time is

            Distance covered for Unit time = d/t

This is the same mathematical equation that we defined for Speed.

As a result finally speed “s” can be easily understood as the distance covered in unit time t and it is denoted as s = d/t.

To check whether the definition of speed fulfills our need to describe translational motions, we should try it with some real life problems, so that we can find its validity.

  

Uses:

As we have already seen, now we can compare the speeds of various runners of various ages using this definition of speed. The shape of the track is not a problem. We can make them run in a circular path and we can evaluate their speed by measuring the perimeter of the path distance “d” and time taken “t”.

Using this definition of speed, we calculated the various speeds of different animals, people, vehicles, and many other natural things.  

And another major use of “d/t” was the measurement of distances.

Using the relation “d/t” we can measure any distance between two places that lies very far away from each other. And we can also predict the time that will take for an object or animal or people having certain speed limits.  

These measurements are often used in travelling and to determine the borders of any kingdom or country.

The measurements were done using the values of Speed, Distance and Time by applying it in simple mathematical operations like multiplication and division.

Let’s start measuring,

We know that  speed = d/t      

Which implies that,     

value of distance d = value of speed * value of time t            

[*Provided that the value of t is not equal to zero as per the conditions of Divisions in Mathematics]

If we know speed “s” and time “t” then distance “d” can be measured.

But at each instant of time, speed can be calculated only if we know length and time.

Then how it can be used the above equation to find distance?  

This problem can be solved by making the speed constant over some time t.

[Note: Constant means nothing but remaining same without any change.  

In physics the changings are happening with respect to time. So if any quantity doesn’t change its value with respect to time then it is called constant with respect to time. In real, all the measurements are done with the help of constancy only. The time of period of this constancy can change but the constancy might happen.

For example we will say the speed of the vehicle is 10 meters per second. From this sentence we can know that the vehicle is achieving the constant speed of reaching 10 meters for one second at some instant. In that instant of small time the vehicle doesn’t change its value.

Though the vehicle changes its speed in the following instants of time, we can be sure that the speed is constant for that instant. Maybe the amount of time being constant can be 1 millisecond or 1 Micro second or 1 Nano second but the constancy happens.

Thus the notion of “being constancy for small intervals or instants of time” is the initialization for limits and it serves as the basic for differentiation and Integration.]

 If speed remains constant for a long time then we can find the distance “d” by using the above equation [d= s*t].

Here if the speed of an object remains constant for 5 seconds with the speed of 5meters per second then we can find the distance covered by that object in 5 seconds by multiplying 5 with the value of speed that is 5 m/s.

It equals to 5 seconds*5 meters/ seconds = 25 meters [using basic mathematics of multiplication and division]

The above calculation hold true only if the object moves in constant speed over time because, in constant speed, the vehicle will reach same amount of length in same amount of time. In the above case, in each second the object will move 5 meters. And for 5 seconds 25 meters.

Though there are no any perfect vehicles or animals that move at constant speed, we can estimate the near value.

Thus constancy helps us to solve many problems.

The simple real life of use of this constancy for a king on protecting his country is…

First we will calculate the speed of a horse or any other vehicle by making it run for some distance. Horse was specified here as just because it is the vehicle for ancient people.

Then it was made to run that horse of known speed constantly over some time t until it reaches the border of the country in the shortest path.

As we have already calculated the value of the speed of the horse, the distance covered by the horse in time t can be calculated as “speed s* time t”.

Thus we calculated the nearly distances of the border of the country from the palace or the capital of the kingdom using the animals like horses.  

So, when there happens a war, we can estimate the minimum value of the time that takes for the enemy troops to reach the palace of the kingdom.

Here we know that “The minimum time to reach the palace is the maximum time got by the king to ready his army”.     

So what is the minimum time?

Speed = Distance / Time

Time = Distance / Speed

If we want Minimum time then

condition 1: the distance (numerator) should be minimum and

condition 2: the speed (denominator) should be maximum as we have studied in mathematics.

[Note: In mathematics, when p/q=r , r is minimum only if p is minimum and q is maximum because p is directly proportional to r and q is indirectly proportional to r]

As because of we have done this experiment in the shortest path, the distance is the minimum one. Accordingly, the first condition is satisfied.  

            The maximum speed can be calculated by taking the fastest horse and making it to run at its maximum speed. According the second condition is also satisfied.

And so, now we can calculate the maximum time the King have to save his palace and his kingdom.   

In the same way, most of the things in transportations can be predicted by calculating the value of speed.

Thus, by calculating speed people solved many real life problems.  

But is it enough?

            Can we tell everything about a motion by just knowing the value of speed?

No.

The definition of motion was further improved by the need of “the direction of the position”.