Exploring the Methods of Differential Calculus through the ...

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Figure 6. Velocity curves for each candidate ramp. Candidate one has the lowest velocity at the start. The velocity of candidate one increases and has the highest ending velocity. Candidate three has the highest starting velocity. The velocity of candidate three decreases and has the lowest ending velocity. There is a point where each pair of candidate ramps have intersecting velocity curves, implying that at one point those two ramps produce the same velocity. In fact, two of the intersections are guaranteed by the Mean Value Theorem because candidate ramp two is a linear approximation between the two endpoints of both candidate ramp one and candidate ramp two. The next step is to find the absolute value of each of the velocity curves. The absolute value of the velocity curve represents the speed induced by the ramp. Once the absolute value is found, the three speeds are compared graphically in Figure 7. Candidate Ramp One: Candidate Ramp Two: Candidate Ramp Three:
Figure 7. Speed curves for the three candidate ramps. The concave up curve starts out with the highest speed but the speed declines and it ends with the lowest speed. On the straight line, the speed is constant. The concave up curve starts with the slowest speed, but the speed increases and it ends with the highest speed. There is a point in this graph at which the curves of the first and third candidates intersect, meaning that they have the same speed at that instant. Also at that point the speed of candidate two is higher than that of candidate one and three. These intersections correspond precisely to the intersections of the velocity curves. After the three speeds are compared the final step is to find the second derivatives of each candidate ramp, representing the acceleration curve of the candidate ramps. The acceleration curves are calculated and shown graphically in Figure 8. Candidate Ramp One: Candidate Ramp Two: Candidate Ramp Three:
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Exploring the Methods of Differential Calculus through the ...

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