In previous lessons, students learned about the importance of using formulas to describe and quantify the motion of an object. These formulas are called kinematic equations. There are various quantities related to the motion of an object-angular velocity, displacement and acceleration, and time. Understanding kinematic equations is essential for the student of physics to predict the behavior of various physical processes. It helps you to solve systems of simultaneous equations, such as those representing the spring system, or to solve for dynamic mass or its equivalent, momentum, in an inertial reference frame.

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Students learn and should master the concepts used in the construction of a simple kinematic equations worksheet. An example is the following. Let us pretend that you were to measure the acceleration of a given mass at one point in time, say, when the object was first made. You would then draw a graph that represents the change in acceleration, or acceleration change, as the mass changed in time, from the initial position.

For each term on the graph, you would write down the corresponding equation, such as the acceleration change at the time t, to the end-effector, velocity change at the time x. The equations can be plotted as a horizontal bar or curved line, or in any other shape desired. The basic functions of the graph, including the parallel chain rule, the forward kinematics rule, and the concept of a second term, which we will discuss later in the lesson, are easily recognized. Once the student has written down an equation for the change in acceleration, he or she may draw a corresponding curve on the graph. This will clearly show the difference between the acceleration change, and the change in velocity, for the term, t, plotted on the graph.

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The second term, z, is often called the displacement function, because it is the function that transforms the acceleration change, from an initial position to an end-effector location. The graph is similar to the first bar in the previous lesson’s Kinematic Equations Worksheet. In this lesson, however, the end-effector is a constant acceleration. To solve this problem, the constant acceleration can be transformed to a set of zero values, using the appropriate function, such as the zero function, tangent function, or Jacobian function.

A final word about the constant acceleration problem, as it applies to the Kinematic Equations Worksheet. Many students forget to include the word “translate” or “apply” when writing an equation for a specific time-step. When they do, the student must either rewrite the sentence in the new time-step, or include a reference page or table that lists the formula for the specific time-step. A better way to teach students to use words like “translate,” “apply,” and “reset” is to provide a clear and concise definition of the concept and then allowing students to either translate the concept directly, or apply it to the appropriate frame of the sentences, depending on whether they are dealing with a constant acceleration, a variable acceleration, or a time-step.

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Exercise Four: Compare your answer for paragraph two, paragraph three, and paragraph four with the answer for lecture five. Assign a grade, and use your class notebook for notes. What kind of relationship does the argument you presented in lecture 5 fit into the model you constructed in lecture 4? How does the model make sense of the data you presented in lecture 5? What kind of relationship does your result support?

Exercise Five: Find your best estimate for the value of the tangent of a curve in terms of the initial velocity V(i). Trace the path of the tangent on the x axis for some arbitrary point on the circle (i.e., midway between two points on the circle). What kind of curve is this? If it’s a polynomial curve, you probably don’t need to do any transformation of the expression; if, however, it’s a quadratic function, it will certainly need to be transformed using the kinematic equations worksheet.

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Exercise Six: Write a short passage using the tangent function V(i) in order to practice problems of the form ax*cos(i+sin(i)+. Plug in your initial condition (i.e., the initial velocity), and solve for the derivative of the function at the points where your tangent function stops. Use the spreadsheet’s function control feature to make sure that the answer you obtain is correct. Using the tangent function on the practice problems kinematics worksheet can help you refine your problem-solving skills.

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