For the following questions, consider the planetary orbit parametrized by r(t) = (Rcos (t³), R sin(1³)) where f(t): R → R is any arbitrary scalar function. (1) Calculate the velocity vector, r' (t), and the acceleration vector, r"(t). (a) Sketch the planetary orbit parameterized by r(t), as well as the vectors r'(2), and r"(2). (2) Write the acceleration vector as a linear combination of r(t) and r' (t). (a) Calculate the projection of the acceleration vector r"(t) in the direction of the velocity vector r' (t). (b) What is the relationship between your answer in part (a) and the speed of the planet? (c) At t = 2, is the speed of the planet increasing, decreasing, or constant? (3) Modify r(t) to find a new parametrization of the curve, such that the planet has decreasing speed at some time value t = to. Sketch r(to), r'(to), and r"(to).

Fig: 1