UNIFORM RECTILINEAR MOTION

In modern times we consider the fastest man to the winner in the final of 100 meters at the Olympic Games. However many people think this is no the correct way to grant this title, and is not accepted unanimously, for example the winner of the 100 meters in Atlanta 96 was Donovan Bailey from Canada, and the winner of 200 and 400 meters was Michael Johnson from United States, they came together to compete in a 150 meters race in 1997, in order to determine who was the world's fastest man.

The discussion by defining the proper distance to grant this title opens the door for a question of mathematical interest:  What do we exactly understand for speed? Well imagine that every sprinter carries a speedometer; the athlete who reports the highest register is the one who gets the title. Although is important to tell that the capacity of every sprinter to maintain a high speed within an interval of time is critical; but we are going to focus on the instant speed that marks the speedometer.

If we don't have a speedometer, How to determine the peak velocity of a sprinter? by the moment we only have the elemental formula: 

distance = velocity x time

That also can be expressed as:

velocity = distance / time

But this formula give as only the average velocity within the interval of time specified and distance chosen, If athlete takes 10 seconds to travel 100 meters, his velocity is 100m/10s  =  10 m/s. But this is not his highest velocity, because in the first meters the athlete strives to reach a high velocity, and in the last meters, he starts to ged tired and maybe his velocity gets low.

So if we calculate his average speed in littler intervals of time, we can have a better approximation of his peak velocity, even more if  intervals are minor, so if we make this intervals approach zero and find limit, we will be able to solve the problem, and will know who's faster. And this brings us back to the concept of the derivative seen before.

Now let’s see again this imagine posted in the article “fundamentals of differential calculus in real variables”.

Imagine while the athlete was running this record was taken:

Time      position

0             0

2s           4m

4s           7m

6s           8.5m

8s           9.2m

10s         9.7m

If we plot the time in the horizontal axis and the position in vertical axis, maybe we obtain a curve in red as follows.

Note the position is not distance traveled, the position is the distance between the particles studied, in this case the particle is an athlete, and a reference point that could be the start of the trajectory.

Now we can take an interval of time t1-t2 as seen in the figure, and for  t1 there is a value in the axis of position x(t1); and for t2 there is another value x(t2).

Then establish the point “P” as [t1,X(t,)] and the point “Q” as [t2, X(t2)].  The slope of the secant line between “P” and “Q” represents the average velocity in the interval of time [t1-t2]. Now let “P” to be stationary and move “Q” along the curve getting closer to “P” reducing the time interval, so at the limit, where the time interval approaches zero, (when Q is so close to P that appear to be the same point) the secant line will become a tangent line, and the tangent line to a curve of time vs position represents the instant velocity, for this case the instant velocity for the time of “P”.

So if we represent the curve in red as a derivable function in a given interval, we’ll be finding the slope of a tangent of the curve at any point that represents the instant velocity in this point we’ve chosen.

That means the derivative of displacement respect the time.

So if we have the equation of displacement as a function of time like this:

X(t) = 9t² – 3t³

It could be easy to find an equation that express the velocity as function of time by deriving the function of displacement, so we have:

v(t) =  dX(t) =  d (9t² – 3t³) = 18t - 9t²

             dt        dt

So we have an expression for velocity as a function of time and we can evaluate it for any time of the traveling.

Note that when velocity is zero in that instant the particle has reached its maximum displacement , and geometrically it can be traduced when the slope of the tangent line is zero, so if we want to know the maximum displacement we have to derive the displacement function and then make it equal to zero, and solve for t, so we’ll having the time for the particle to reach its maximum displacement, and evaluating that time in the equation of displacement we will be finding the maximum displacement.