Rocket Position and Velocity

Date: 04/12/2001 at 08:52:30
From: Asha K.R.
Subject: Trapezoidal as well as Simpson's rules

A rocket is launched from the ground. Its acceleration, measured every 
5 seconds, is tabulated below. Find the velocity and position of the 
rocket at t = 40 seconds. Use trapezoidal as well as Simpson's rules. 
Compare the answers:

  t      0     5      10     15     20     25    30    35    40
  u(t)  40.0  45.25  48.50  51.25  54.35  59.48 61.5  64.3  68.7

Date: 04/14/2001 at 13:48:34
From: Doctor Jaffee
Subject: Re: Trapezoidal as well as Simpson's rules

Hi Asha K.R.,

If you were to plot the points (t, u(t)) on a grid, connect the points 
(0, 40) and (5, 45.25) with a line segment, and then draw a vertical 
segment from each of those points to the t-axis, you would have 
constructed a trapezoid. Use the same method to construct trapezoids 
between each pair of adjacent points and you will end up with eight 
trapezoids, the sum of whose areas approximates the area under the 
implied curve that connects the nine points.

Since the velocity of a function in the first quadrant is the 
anti-derivative of the acceleration (the area under the curve), the 
total area will tell you the velocity after 40 seconds.

The shortcut for finding the sum of all eight trapezoids is the 
trapezoid rule, which states that the sum is:

     b - a
     ----- [f(a) + 2f(x1) + 2f(x2) + ... + 2f(x7) + f(b)]
       n

where a = 0, b = 40, n = 8, and each of the xn's corresponds to one of 
the t values.

Simpson's Rule is similar except that is based on taking the points 
three at a time and fitting parabolas around them. The formula is:

  b - a
  -----[f(a) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4)+ ... + 4f(x7) + f(b)]
   3n

Perform the computations and you should get a result pretty close to 
the value you got with the trapezoid rule. Normally, Simpson's Rule is 
a little more accurate given the same number of intervals.

Now, if you compute the velocity for each 5-second period, you can 
create a new set of data whose anti-derivative will provide you the 
position values.

Give it a try and if you want to check your answer with me, write 
back. If you are having difficulties, let me know and I'll try to help 
you some more.

Good luck.

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/