Because Lines Don’t Curve – Hooke’s Law and Quadratic Euler’s Method Question: Given an initial stretch, how can we predict the position and velocity of a mass attached to a spring at any future time? Audience: This investigation is appropriate for BC Calculus students who are familiar with tangent lines and linear approximation. They should be comfortable using Euler’s Method on simpler problems, but they do not need to be able to integrate. I find the most valuable aspect of this investigation is motivation for the discovery of Taylor polynomials, which leads naturally to the development of Taylor series. Process: The first goal for the students is to write a differential equation – but this may take some guidance. Experience tells us that when you pull on a spring, you feel the spring pull back against you. When you stretch it farther it pulls back with greater force. Hooke’s Law states that the force exerted on an object attached to an ideal spring is in the opposite direction of, and in proportion to the amount the spring is stretched from its equilibrium position. This gives: · Newton’s 2nd Law tells us (among other things) that a force exerted on a mass causes it to accelerate. · Together Hooke and Newton tell us: ·
· And we arrive at our first goal, the differential equation: · Presumably, the students can’t yet integrate, but even if they know how to solve separable differential equations, it is not so easy to see how to rewrite this as a separable D.E. Now that we have a differential equation we are ready our next goal ‐ writing Euler’s Method equations. Δ
0 Δ
0 Δ
Using ,
1, Δ
0.05, we obtain the following graphs: 6
6
4
2
2
0
0
‐15
‐2
Displacement vs. Time 5
25
45
65
4
‐15
‐2
‐4
‐4
‐6
‐6
Velocity vs. Time 5
25
45
65
As we can see in the above graphs, Euler’s Method does not provide an accurate solution to our problem. So we turn to Quadratic Euler’s Method. The general quadratic Euler’s Method equations for generating ordered pairs , are: Δ ·
Δ
,
Δ
,
As mentioned in the TCM presentation, the formula for comes from the equation for a second degree Taylor polynomial. For the Hooke’s Law equations, the second derivatives are: Thus our quadratic Euler’s Method equations are: Δ
0 Δ
Δ
0 Δ
Δ
1, Δ
,
Using 0.05, we obtain the following graphs: 2
2
Displacement vs. Time ‐15
Velocity vs. Time 1
1
0
0
‐1
5
25
45
65
‐15
‐2
‐1
5
25
45
65
5
25
45
65
‐2
4 Comparison of Linear and Quadratic Euler’s Method: 6
6
4
4
2
2
0
0
‐15
‐2
5
25
45
65
‐15
‐2
‐4
‐4
‐6
‐6
Adapted from: Bartkovich et al. Contemporary Calculus through Applications. Dedham: Janson Publications Inc., 1996.