Question

4 Rally-car Tracking [5 marks] You are testing software designed to track race cars as they race around a track, to improve the quality of camera-work in race broadcasts on television. You are in a blimp, high above a speedway. The cabirn of the blimp has a glass floor, and by looking down, you can observe the racetrack below. There is a cair doing a time-trial. The software produces the following parametric equation approximating the position of the car at any given time, t. You may assume a, b, m and n > 0 are constants. a cos(mt) y(t)bsin(nt) r(t) (a) Find the velocity vector and acceleration vector of the car [2 marks] (b) Calculate the unit vector tangent to the trajectory of the car (call it τ(t), and using T(t), determine a unit vector normal (perpendicular) to the trajectory of the car (call it n(t)) [1 mark] (c) Acceleration can be defined by its tangential and normal components to the trajectory. Tangential acceleration (which we will denote at) determines the propulsion of the car. Normal acceleration (which we will denote an) describes the turning of the car In mathematical notation, this means that you could express acceleration as a(t) = atr(t) + ann(t) (i) Using your results from (a,b), decompose the acceleration vector into tangential and normal compo- nents at and an using scalar products (you do not need to attempt to simplify the result) [1 mark] (ii) An object undergoing uniform circular motion travels at a constant speed along a circular path. Assuming m, n E (0.27) the vector r(t) describes uniform circular motion when a = b and m n. Show that under these conditions the tangential acceleration is 0 and the normal acceleration is given

Answer 1

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