1. Consider the ellipse given by the equation
(((x − 1)^2)/9)+(((y + 2)^2)/25)
(a) Sketch this ellipse in the x-y plane. (You do not need to label intercepts, but other important features should be labelled.)
(b) Particle A starts at the rightmost point of the ellipse and performs a circuit in the anti- clockwise direction every 2π seconds. Give a set of parametric equations describing the x- and y-coordinates of A as a function of time t.
(c) Do the same for particle B, which starts at the leftmost point on the ellipse and performs a circuit in the clockwise direction every π seconds. Explain the differences between this and your answer to (b).
2. Bames’ brother Bamie is playing in an AFL game.1 He kicks the ball from position O = (0, 0, 0) towards the goal. The bases of the goalposts are at (40, 10, 0) and (40, 16.4, 0). The ball follows a path described by the parametric equations (x(t), y(t), z(t)) = (10t, 3t, (8√5t − 4t^2)) (where t is in seconds), until it hits the ground, at which point we will assume for simplicity that it stops and does not bounce.
(a) At what time does the ball hit the ground?
(b) If the ball passes between the goalposts, what must its x-coordinate be at that moment?
What conditions must the time t and y-coordinate satisfy?
(c) In the absence of any opposing players who might have been able to stop the ball, did
Bamie score a goal?
(d) Sketch a graph of the path the ball followed, as seen from above, i.e. in the x-y plane. Indicate the positions of the goalposts, and label the important features of the graph.