Charles B. Wakefield
Department of Science, Mathematics, and Computer Science
University of Texas of the Permian Basin
Odessa, Texas 79762


In this study, using the computer algebra system Mathematica, we consider the evaluation of the velocity at any point along an irregular pathway. We assume there is friction which is a function of the horizontal distance along the pathway. Since there is friction, we can't assume conservative forces. Also, because the pathway is not simple we must use numerical integration to evaluate the distance traveled. This is necessary because for nonconservative forces, the force is dependent on the path.

The pathway data is fit with a fifth degree polynomial. The distance along the path is calculated with the numerical integrator for several different values of the independent variable. These distances are plotted and then fitted with a cubic fitting function. This gives us a function which will allow us to calculate the distance along the path at any point on the path. Using the work energy relation for nonconservative forces[Graphics:phy_h~1.gif]


we derive an expression for the final velocity as a function of x. This [Graphics:phy_h~3.gif] is then plotted for the region of interest. We then show a graphic which illustrates how the velocity will change with a range of different values of the coefficient of friction.

The problem is understandable to students but not doable without a Computer Algebra System. This problem illustrates vividly the Calculus in Calculus Based Engineering Physics. It can be used to illustrate the use of a computer algebra system and the basic Calculus involved in the Engineering Physics course and the Calculus II course.

Load needed packages.


Load the irregualr pathway data.


Find the fitting function for the pathway data and show the plot of the function and the data so we can see how well the fitting function fits the data.




Note the fitting function, a fifth degree polynomial, fits the irregular pathway data well.


Fig. 1 Fitting function for irregular pathway

Calculate the derivative of the fitting function and the square of the derivative for use in calculating the distance along the pathway.


The distance along the pathway must be determined by numerical integration, since the expression for the arc length can't be integrated in closed form. So, we build a table of values which we will plot and then fit with a polynomial. This will provide us with a smooth function for determining the distance along the curved pathway.



Determine the fitting function for distance along the pathway.



Note in the following graph that the fitting function dist(x) fits the pathway distance data very well.


Fig. 2 Fitting function for distances along the pathway

Set up the input data, initial values etc.


Now we are ready to write down the equation for the velocity at any point along the pathway from x = 1 to x = 3. We will then check one point by letting the friction [Graphics:phy_h~25.gif] be equal to zero and use the fact that without the dissipative force of friction the problem should collapse to the simpler conservative force problem where [Graphics:phy_h~26.gif]. This calculation gets exactly what we get when [Graphics:phy_h~27.gif] is zero. When we increase [Graphics:phy_h~28.gif] the velocity slows down as we would expect.


From the work energy theorem for non conservative forces we know that the following is true.

(1) [Graphics:phy_h~29.gif]

In this equation [Graphics:phy_h~30.gif]. In this problem [Graphics:phy_h~31.gif] changes with x. However, the derivative at any point along the curve gives the slope of the curve at that value of x. This slope is equal to the tangent of the angle, [Graphics:phy_h~32.gif], makes with the x axis. So, [Graphics:phy_h~33.gif]. Therefore,

(2) [Graphics:phy_h~34.gif]

If we substitute this result into equation (1) and recall that [Graphics:phy_h~35.gif], we have:

If this equation is solved for [Graphics:phy_h~37.gif] we arrive at the final working equation.

(3) [Graphics:phy_h~38.gif]

For out first calculation, we will use [Graphics:phy_h~39.gif].


Using the working equation above turn off the friction and let us see if the results gives us the value for a conservative force system which is evaluated with the "Solve" instruction.



Now let us evaluate the velocity when some friction is introduced. We see from this result that the friction has slowed down the magnitude of the velocity.


Plot vf for sigma = .04 over the entire domain of interest, that is x = 1, to x = 3.



Fig. 3 Velocity profile with friction included. In this case [Graphics:phy_h~50.gif], the horizontal distance.

We will next assume that the friction [Graphics:phy_h~51.gif], the actual distance along the irregular path rather than x, the horizontal distance. The plot is supressed and will be shown later.




Use the function of [Graphics:phy_h~55.gif], the actual path distance squared. The plot is supressed and will be shown later.




In the next plot we see all three functions tried for the friction [Graphics:phy_h~59.gif], the green curve, [Graphics:phy_h~60.gif], the blue curve, and finally, [Graphics:phy_h~61.gif], the red curve.


Fig. 4 This figure shows the velocity profiles over the irregular path for three different functional relationships between [Graphics:phy_h~64.gif] and the path. Green is the horizontal relation, blue is the actual path way function and red is the squared path way data. The particle will stop if [Graphics:phy_h~65.gif] is increased between x = 2.7 and x = 3 for the red curve.

Compare the velocity profile above with the actual path in the following graphics array.


Fig. 5 Comparing the two we see that the velocity rises as the path decends and vice versa.

Change vf so that we can plot the same graph with several different values of friction.



In the following plot the top curve is the velocity for the friction turned off. The lower curves gradually show the slowing down as the friction increases from zero to .3. In each case we assume that friction is constant throughout.


Fig. 6 Velocity profiles with the sigma varying from 0.0 to 0.3. The yellow curve is the sigma 0.0 and the purple curve is the sigma = to .3.

In conclusion, we have shown how to calculate velocity profiles over an irregular pathway using several different functional relations for the friction coefficient.


1. Raymond A. Serway, Physics For Scientests & Engineers with Modern Physics, Third ed., Updated Version, Saunders Golden Sunburst Series, 1992