AN INTRODUCTION TO NEWTONIAN MECHANICS
by Edward Kluk
Dickinson State University, Dickinson ND

FREE FALL MOTION

        Why and how to make a simulated experiment of free fall motion
        Suppose you have never study any Physics before. Your first assignment is to investigate and describe a free fall motion. Such motion occures if a body is lifted up, stopped, and let to fall down with no significant air resistance. In 16th century Galilei Galileo investigated free fall motion and concluded that all bodies fall down identically. Good examples of it are falls of metal or glass solid balls, as opposed to a sheet of paper which is initially in a horizontal position. Even taking a styrofoam ball you may notice an air resistance comparing its fall with a fall of solid metal ball. An obvious qualitative observation for motion of all mentioned above balls and many other compact objects is a gradual increase in their speed in the initial phase of the motion. Taking quantitative data, like covered distances for different times, is more difficult. Photogates, ultrasonic rangers or other suitable technologies are needed. Human beings have too slow reflex for that kind of measurements on the surface of our planet. This is only real obstacle because nowadays almost every wrist watch with digital display has a stopwatch of accuracy of 0.01 s. Our reflex, however, has accuracy of about 0.1 - 0.2 s, and this is not good enough to make meaninfull measurements. Now you should comprehend why ancient and medival scientists had difficulties with understanding even simple motions. They were deprived of reliable experimental data.
        If you have not enough money to buy one of the mentioned above technologies, but you still have good enough computer, imagine yourself a different planet with much lower gravitational pull. On such planet a free fall would be much slower and our reflex would not be an obstacle in taking reliable experimental data. This applet provides you with simulation of the planet. When you click on Start button the red object blinks and falls freely 10 m down. Do not try to stop it. When the object reaches the bottom, you can reset the applet and start all over again. Now measure with a stopwatch and record times the object needs to reach the levels 1 m, 2 m, ... , 10 m from the level 0 m. Remember to record your data in a neat fashon, and whatever you will be doing with them, do it rather precisely. Too much of precision (precision beyond of exerimental errors) if you are aware of such possibility is usually not as dangerous in science as not enough of it.

       Interpretation of the experimental data
        If you are a math wizard and you have any curve fitting programs try to fit simple curves to the collected distance versus time data. This way you will rediscover the law of free fall. If you do not feel too comfortable with math, do not worry. Neither Isaac Newton nor Albert Einstein knew enough math for their purposes. Newton created math he needed (calculus) and Einstein learned it (Riemann geometry) in "on a job training". Make a neat distance versus time graph and have a good look at it. Well, it looks like a curve. But what kind of curve? Here is a hint. Graph the distance versus square of time and you should be able to fit to your data a straight line running through the origin (0,0) of the graph. If so, then you have rediscovered the free fall law stating that the covered distance is proportional to the square of time. It means that the graph distance versus time must be a simple parabola with vertex at the origin.

        Math helps to reach more conclusions
        Making the measurements you have not had any problems with ideas of time and distance. You have learned how to measure them very early, and most probably with no reference to any science. But be aware, knowing how to measure something is not equivalent with full understanding of what this someething really is. To measure something we need an operational definition which is an instruction how to measure it. To understand it much more is needed. You may have heard that the Big Bang or begining of our Universe (according to the Big Bang model of the Universe) was a begining of time and space. It sounds simple, but it is not simple at all. in other words we are not quite sure what time and space are.
        At least intuitively you understand what is a speed. In this country we deal with it practically every day driving our cars. If you drive your car with the steady speed 70 mph along a long streach of Interstate it means that in 1 hour it covers 70 miles, in 1/2 of an hour 35 miles, in 6 minutes just 7 miles, and so on. What can we say about the speed of our freely falling object? You know from our experiment that the distance it covers is proportional to the square of elapsed time. Moreover, you know the proportionality coefficient because it is equal to the slope on the graph of distance versus square of time. Calculating the slope, find a "distance" along the distance axis in meters (rise) as they are marked there and a related to it "distance" along square of time axis in seconds squared (run). Find rise/run ratio. Marking this ratio as a/2 , covered distance by d and elapsed time by t, you can write the rediscovered law of free fall as a simple mathematical relation
d = a t 2 / 2 .
Now we need a definition of speed. If we take two distances d 1 (shorter) and d 2 (longer) and related to them two elapsed times t 1 and t 2, then it make sense to define an average speed in the time interval ( t 1, t 2) as
vav = ( d 2 - d 1 ) / (t 2 - t 1 ) .
This definition is valid for any dependence of d on t . In our specific case d is a quadratic function of t. Therefore substituting d 1 and d 2 in the second equation with help of the first equation we obtain
vav = (a / 2) ( t 22 - t 12 ) / (t 2 - t 1 ) = (a / 2) (t 2 + t 1 ) .
Here for final simplification well known algebraic identity a 2 - b 2 = (a - b)(a + b) was used.
        If the time interval ( t 1, t 2) is very short, then t 1 and t 2 can be replaced by current time t, and instead of an average speed we have an instant speed v at time t v = a t .
You should notice that starting with experimental dependence of covered distance on time and doing some elementary algebra, we have proven a linear dependency of a free falling body speed on time. The constant a there represents a rate of change of speed or body's acceleration. The acceleration for a free fall is usually marked as g , and on the Earth surface is equal to 9.8 m/s2. On our fictitious planet as you already know g is much smaller.
        Moreover, introducing the idea of instand speed we have applied a method which is used in mathematics to introduce a function derivative. As a matter of fact if you take derivative of d with respect of t the expression for an instant speed follows instantly. This should convinced you that studying physics it is impossible to avoid mathematics. People who propagate "a conceptual physics" as physics without math should read what one of the best physics theorists Richard P. Feynman have said about it.

        Evaluation
        If at this point you do understand:

the objectives of this lesson are fully achieved. If you have doubts try to read it once more concentrating on them, but do not try to memorize this text. physics is not about memorizing, it is about understanding.

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Last update: Jan 10, 1997 E - mail to Edward Kluk
Copyright (c) 1996 Edward Kluk