Problem 3: [20 points] An Application of Taylor Series in classical [Newtonian] mochanjrs, the kinetic energy, K156, of a nonrotating rigid body depends on the mass of the body and its speed. It can be calculated using the formula KEG = (1/2)mo2, where m is the mass of the object and o is the object's speed. In the theory of special relativity, the total energy of such an object takes into account its rest energy as well as its kinetic energy and is given by the formula El?!) 2 L32, 2 1 (1;. ) where c is the speed of light. 1 A. [12 pts] Show that the fourth order Taylor polynomial for Eh!) centered at o = O is sort-c.2 + Emu2 + shy? Hint: There an: many ways to approach this problem, some of which are more efcient than others. Think about the various ways we have discussed nding oocicicats for Taylor polynomials. Regardless of how you approach this pmblem, you must show all of your rolctdatloas by hand (but, as always, you are welcorm: to use technology to check yam~ mulls). Solutions to. Match emissary calculations am done only by technology will woe-too no more than. 3/12. (Problem 3, continued) B. [2 pts] The rest energy of the object is given by E, = me , so the total relativistic kinetic energy, KE,(v) is given by KEr(v) = E(v) - mc. Using the fourth order Taylor polynomial you found in Part I, write down the fourth order Taylor polynomial for the KE,(v) centered at v = 0. C. [6 pts] Most commercial jets have a mass of about 80000 kg and reach a cruising speed of approximately 250 m/s. Given that c = 3 x 10# m/s, do the following. i. [2 pts] Approximate the relativistic kinetic energy by using its fourth degree Taylor polynomial found in Part B for v = 250m/s. Report your answer to 1 decimal place. Do not list your answer using scientific notation. ii. [1 pt] Calculate the classical kinetic energy when v = 250 m/s. iii. [3 pts] Use your results from Parts i and ii to explain whether special relativistic considerations are important to take into account in this scenario