Rocket Position and VelocityDate: 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/ |
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