Jimmy and Doraemons have built a small rocket that carries fireworks for the Celebration planned for the opening of the UNSW Mars Campus. The rocket has mass m > 0 and is fired from a launch pad on the Village Green. The horizontal distance of the rocket from the launch pad at time is () and the height of the rocket above the Village Green is y(t), so the position of the rocket is given by the vector r(t) = x(t)i + y(t)j. At the time of launch, t = 0, z(0) = 0 and y(0) = 0. The forces acting on the rocket are a constant gravitational force with magnitude mg acting downwards and atmospheric resistance which, in the very thin atmosphere of Mars, is proportional to the velocity v(t) = 2' )i+ (8)j. You are given that the equation for y(t) is: my" = mg kyf: (1) where k > 0 is a constant. (a) Choose the correct equation for m(t) from the list below: ma' = ka! - (b) Choose all the terms below that correctly describe the differential equation (1) for y(t): homogeneous separable inhomogeneous linear second order [ first order (c) Find the roots of the characteristic equation corresponding to equation (1) and enter them in the box below. If there is more than one root, separate the roots with a comma. 0, -k/m & (d) Consider (1) when m = 36 and k = 2.4. The general solution of (1) with the initial conditions y(0) = 0 and 3'(0) = V can be written: y(t) = 8(V +d9) (1 - exp(Bt)) dgt. Enter the value of the constants 3 and & as exact values or correct to at least 8 significant figures. Syntax advice: Enter your answers as exact values or correct to at least 8 significant figures. Eg, 45/83 or 0.54216867. (e) Jimmy and Doraemons think it will look best if the fireworks carried on the rocket detach and explode when the rocket is at its highest point from the Village Green. If we take V' = 6 mg/k, and write the time at the highest point ast = mT/k, what is the number T'? e B