Jacky Chong - Jacky Chong IB Student D1 HL Mathematics...

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Jacky Chong IB Student D1 HL Mathematics Internal Assessment Type II December 14, 2009 Internal Assessment – Designing a Freight Elevator In this paper, I will analyze a given model for the motion of an elevator used to carry freight in a mine. More importantly, I will evaluate its strengths and weaknesses and create my own model with specifications. Part I: Analysis of Given Model I.a Variables, Parameters and Constraints of Given Model The function of the given model is . ” is representing time (in minutes) and “ ” is the position of the elevator. = 0 means that the elevator is at ground level, and = 0 represents the starting time. Therefore, the function tells how the elevator’s position changes in terms of the change of time. Firstly, let us talk about the variables. Variable means values that change. There are a few variables in this situation: time, position, velocity, acceleration and change in rate of acceleration, of which we rarely explore in both Mathematics and Physics. Secondly, let us talk about the parameter. Parameter is an essential element of parametric equation. When we have a function describing a 2 dimensional motion, for example, , where both “ ” and “
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describe the position of an object, we have a path of the object’s movement. Figure 1 : , path of the object, described with coordinates, both and are position variables. However, we do not know its moving direction, speed and acceleration. Therefore, we add a parameter to the function, which means to make both “ ” and “ ” be in terms of “ ” – time. We let be , and be .
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Figure 2 : Parametric function of in tracing dot form with 0.1 increment: and . There is nothing left to the y-axis because when , and is the starting time. Let time be measured in seconds. The arrow shows the motion’s direction. Now, we know the direction, speed and acceleration (both one minute, but , so accelerating) of the object’s movement.
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Figure 3 : Parametric function of in tracing dot form with 0.1 increment: and . Path and direction is the same as the graph in Figure 2, but distances between dots increase more violently than the graph in Figure 2, which means greater acceleration and speed. We can also let be and be . The path and direction is the same, but speed and acceleration will increase. Now, we see how parameter and parametric equation help us to add a third dimension (time) to a two dimensional graph (position) with only two axes, and let us know the direction, speed and acceleration. However, in the elevator case, the elevator only goes up and down, which is an one dimensional movement. It only needs one
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axis, so we can let the other axis be time. In this situation, we already know the direction, speed and acceleration of the object with single equation function, so we don’t need to use parametric function. Therefore, throughout this paper, I will not apply parametric function. However, There is still two parameters in the function – “ ”, time; and “ ” or “ ”, position. They are parameters because they are the same for one graph, but you get a new graph if you change them.
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