Parabolic Path III: An object moves along the parabola y = x2. At various times, t,the

Chapter 10, Problem 9

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Parabolic Path III: An object moves along the parabola y = x2. At various times, t,the object is at various points, (x, y), where xand y are in centimeters and t is in seconds.a. Write the position vector (x) as a functionof x alone (and the two unit vectors and ,of course!). Then find the velocity vector(x) as a function of x and dx/dt.b. Assume that the object moves in such away that x decreases at a constant rateof 3 cm/s. Find (2). How fast is the objectmoving when x = 2?c. Sketch the graph of the parabola and draw(2) and (2) at the point (2, 4). Explain whythe graph of (2) is reasonable.d. Find the acceleration vector, (x), andevaluate (2). Sketch (2) on your graph.e. Find the tangential and normal componentsof acceleration at x = 2. Show thesecomponents on your graph. Based on thegraphs, why are your answers reasonable?f. When x = 2, is the object speeding up orslowing down? Justify your answer.g. The object changes its motion and goes insuch a way that its speed along its curvedpath is 5 cm/min. Write an expression interms of x for dL, the differential of arclength along the curve. Find dx/dtwhen x = 2.