Example 08
Series, Limits, Numerical Determination of Roots of Equations, and Numerical Integration

Copyright © 1993 by Bill Titus, Carleton College, 
Department of Physics and Astronomy, Northfield, MN  55057-4025
September 6, 1993

Topics and Skills

Problem

Solution - Part a

Step 1 - Using  Integrate[], complete the following code that defines  a,  v,  and  x  as functions of  c  and  t.

Clear[c, g, xo, a, v, x, t, s]
a[c_, t_] = -g Exp[-c t]
v[c_, t_] =
x[c_, t_] = xo +

Comment 1

Solution - Part b

Step 1 - Try to evaluate your functions at  c = 0.  Describe what happens.

Comment 1

Step 2 - You can find the limiting results using a MMA function.  Any guess what it's called?  Use Appendix 8 in Wolfram to find out about this command and then apply the command to determine the three desired limits.

Step 3 - Some physicists would find these limits by doing Taylor Series expansions on the functions about  c = 0.   Find out information about the MMA command  Series[]  and apply it to determine the Taylor Series expansion of  x[c, t] to third order.  Name the latter expression  xSeries  so you can refer to it later.

Comment 3

Step 4 - For  c = 0.5,   xo = 100,  and  g = 9.8,  plot the third order approximation to  x[c, t]  and the corresponding exact result on the same graph.  To convert  xSeries  into a function, you must first remove the order term.  To do so, use the MMA command  Normal[].  Find out about this command and use it to produce a function  xApprox[c, t] which gives the desired third order approximation.  Then create the plot over the time interval between  t = 0.0  and  t = 8.0.

Comment 4

Solution - Part c

Step 1 - Try to use  Solve[]  to find the time  to  at which Sarah strikes the ground.  Be sure to use  Clear[g, xo, c, to]  before you invoke  Solve[].  Why?   What do you find?

Comment 1

Step 2 - Assume c = 0.5,  xo = 100,  and  g = 9.8.  First make a plot of  x[c, t]  to estimate the time  to  where  x[to]  is zero.  What do you find?

Comment 2

Step 3 -  You can improve your estimation by numerically determining the root of the equation  x[0.5, t] == 0.  The MMA command to do this numerical calculation is  FindRoot[].  Find out information about this command and use it to determine  to.  Call the result  myRoot.  Then find the values for  x[to]  and  v[to]  using replacement rules.

Comment 3

Solution - Part d

Step 1 - Suppose Sarah's acceleration is given by  -g cos[sin[c t]]  with  c = 0.5,  xo = 100, and  g = 9.8.  Cut and paste the equations from Step 1 of Part a.  Change the expression for the acceleration to the desired form and activate the cell.  What the output implying?

Comment 1

Step 2 - Get information of the MMA command  NIntegrate[].  What do you find?  Apply this function to determine the following integrals

1/(1 + Exp[x])         from  0  to  Infinite
Exp[Cos[x]]            from  0  to  2 Pi
Sin[x] BesselJ[0, x]   from  0  to  1

Step 3 -  The following code defines  v  and  x  in terms of   NIntegrate[].  Look over the code and then activate the cell.

Clear[c, g, xo, a, v, x, t, s]
a[c_, t_] = -g Cos[Sin[c t]];
v[c_, t_] := NIntegrate[a[c, s], {s, 0, t}]
x[c_, t_] := xo + NIntegrate[v[c, s], {s, 0, t}]

Comment 3

Step 4 -  Use these function to plot Sarah's velocity and position from  t = 0.0  and  t = 8.0  for the parameters  g = 9.8,  xo = 100, and  c = 0.5.  Be sure to label the axes on your plot.

Comment 4

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