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

HORIZONTAL PROJECTILE MOTION


        A technical introduction
        This applet simulates all kinds of projectile motion with low gravity acceleration and makes possible to time these motions. A motion can start from eleven marked heights at the distance 0 m. Its initial speed and direction are selectable from three labelled choice boxes. If default sign "+" is selected, a selected projection angle counts counter clockwise from a dotted horizontal line. Otherwise it counts clockwise. Every time the red object reaches the edge of the vision field it stops and Reset button is activated. This button let us restore initial conditions for the last motion. The clear button restores default initial conditions for the motion and repaints the vision field. Finally, if the path tracer is on, a path of the object is drawn in the vision field.

        How vertical and horizontal motions add
        We already know mathematical models for free fall and undisturbed motion on a frictionless and impenetrable horizontal plane that eliminates a gravitational pull . A natural question arises what will happen if a body starts with nonzero horizontal speed without a support of the horizontal impenetrable plane. A simple experiment with a body sliding with some horizontal speed off a table shows the body motion as a combination of vertical and horizontal motions. As we already know an accurate timing of this kind of motion is not possible without high tech devices like a fast photography. Some camcorders have fixed time between consecutive frames and they can be used to study this kind of motion. But even with such camcorder our experiment would be complicated.
        It is much simpler to go back to our fictitious planet with the very low gravity acceleration simulated in the applet. Start it with all default settings except of speed which should be set at 0.20 m/s. This will simulate a case of the motion of our current interest. Measure all time intervals the moving object needs to reach each consecutive vertical dotted line from its starting point. These consecutive dotted lines are in a distance of 1 m from each other. Therefore in the first measured time interval the object covers 1 m in horizontal direction , in the second interval 2 m, and so for. A total distance covered by the object for each interval is greater than that because it moves simultaneously in both horizontal and vertical directions. Graphing horizontal distance versus time you will get exactly the same result as for the motion along a horizontal frictionless plane. Namely, the horizontal component of object's speed vH remains constant because your graph is a straight line. The slope of this line represents the mentioned above horizontal component of the speed. Let us introduce a two dimensional Cartesian frame of reference with an origin at zero height and zero displacement which is in the lower left corner of the vision field, x axis directed horizontally to the right and y axis directed vertically up. Then, you should be able to conclude from your graph that the elapsed time t and x coordinate of the object in the Cartesian reference frame are related by the following formula

x = vH t ,

representing a uniform motion (motion with a constant speed and no change of direction) along x axis.
        Now it is time to collect next set of data related to change of object's vertical position. Measure all time intervals the moving object needs to reach each consecutive horizontal dotted line from its starting point. These consecutive dotted lines are also in a distance of 1 m from each other. Relying on your experience with a free fall motion you may expect that a covered vertical distance d may be proportional to t 2 rather than t. Graph d versus t 2 looking for "experimental" confirmation of this hypothesis. Your results including value of acceleration should be practically identical like for the free fall case. After all you are on the same low gravity acceleration planet. Therefore

d = a t 2 / 2


where the acceleration a should be about 0.01 m / s 2.

        Math helps to reach more conclusions
        The vertical distance d covered by the falling object is related to its y coordinate. If an initial y coordinate is denoted as yo then

y = yo - d = yo - a t 2 / 2 .


Mathematical forms of x and y as functions of time deduced from our "experimental" results show that the motion of the object is composed of two independent simple motions. The horizontal motion with a constant speed vH and vertical free fall with a constant acceleration a. They are independent in this sense that a modification of the horizontal motion by changing its initial speed vH does not influence the vertical motion and a modification of the vertical motion by changing its acceleration does not influence the horizontal motion. To check "experimentally" the first of these two properties of this kind of motion, make timing for a vertical component of motions with initial horizontal speeds 0.1 m / s and 0.15 m / s. If horizontal and vertical components of these motions are independent then the vertical timings for both selected motions should be the same within of an experimental accuracy. You should be aware that if an air resistance plays an important role the horizontal and vertical components of motion are not independent anymore.
        Calculating t from the formula for x and substituting the result to the formula for y gives the following result

y = yo - a (x / vH ) 2 / 2 = yo - a /(2 vH2 ) x 2


In the defined above Cartesian reference frame this relation formally describes a parabola with opening down and vertex at x = 0 and y = yo. The upper part of the right branch of this parabola describes the path of the object. To visualize it,  run the applet with the path tracer on and other parameters as defaults. Surprisingly enough, Galileo knew that paths for this kinds of motion are parabolic. Now insert into the parabola equation for yo and vH their default values 10 m and 0.25 m / s . Select a few simple values for x and calculate respective y values comparing them with values for the same x values along the trace left by the object. If our mathematical model of horizontal projectile motion is correct the calculated and "experimental" y values should be in a good agreement.

        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.



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