ORBIT ved www.systime.dkMotion - instantaneous velocity


In this section You will learn how One (to a certain degree) can determine the instantaneous velocity of a body in motion.




One more time we will examine the situation with the boat. In order to survey the motion of the boat we will measure the associated values of time and position. The quality of this study depends on the number of measurements.

Start example1. Every 4'th second we measure the position and plot the point into the (t,s)-graph. Using the graph we are able to in each time interval to calculate the average velocity as being the geometric slope of the secant during this time interval. 

Example2: Dt = 2 s.    Average velocity.
Example3: Dt = 1 s.    Average velocity.
Example4: Dt = 0.5 s.    Average velocity.
Example5: Dt = 0.25 s.    Average velocity.
Example6: Dt = 0.1 s.    Average velocity.

As You see, we are able to get a more and more accurate impression of the graph of the velocity by measuring in smaller and smaller time intervals Dt.



We have just been shown, that when we make measurements in smaller and smaller time intervals the average velocity vil get closer to the 'real' velocity - that is the velocity carried by the body at some moment t. We denote this velocity the instantaneous velocity v(t).

Let's take a closer look at the measured velocity of the boat as it passes two position marks. One of the position marks is placed right where the boat passes after exacty 4 seconds. The other position mark is placed to the right of the first mark. We wish to determine v(4 s).

In the examples beneath the average velocity is calculated as the slope of the secant during the specific time interval.

Example1: Dt = 4 s. Show the secant.
Example2: Dt = 2 s. Show the secant.
Example3: Dt = 1 s. Show the secant.
Example4: Dt = 0.5 s. Show the secant.
Example5: Dt = 0.2 s. Show the secant.
Example6: Dt = 0.1 s. Show the secant.

Right-click the (t,s)-graph and maximise the new window.
As the graph shows You, the secant will get closer to the tangent of the graph at the point
t = 4s, when  Dt gets small.

We end up with a measured average velocity of 1.89 m/s, which is rather close to the instantaneous velocity of 1.85 m/s at the point.


The conclusion on the above exercises is right at hand:

thus  v(t) = atangent.

The (t,s)-graph of a motion is the graph of the function of position s(t). From a mathematical point of view the slope of the tangent atangent also depicts the derivative at the point, thus

s'(t) = atangent     which leads to     v(t) = s'(t).


In the animation given above the position as function of time was programmed as

s(t) = ½.ko.exp(½.k.t),    where  k= 1 s-1 og  k0 = 1 m ,

thus leading to an instantaneous velocity (or just velocity) of

v(t) = s'(t) = 1/4.ko.k.exp(½.k.t),

By insertion we get

v(4 s) = 1/4 m/s.exp(½.1 s-1.4 s) = 1.85 m/s.


Final remark:

If You know the function of position s(t) related to the motion of a body, all You have to do is to find the derivative of this function in order to achieve the function of velocity. In principle You now have all the information worth knowing about the velocity.

The laws of nature is based on mathematical formulas and functions, but unfortunately we are not always able to get these in advance. Thats why we have to do experiments and measurements in order to wrest some secrets from nature.

In this connexion it is important to realize that it is not possible to measure the velocity in a point! This is solely a mathematical puzzle. We neccessarily have to perform two measurements surrounding the point in which we wish to determine the velocity. The closer the measurents the closer we get to the instantaneous velocity. Practically, the limits for this process depends on the equipment available.

In the following we will walk through a method bringing up the determination of the function of velocity.


 - sorry, not translated.

Move on to Motion - speed

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