It is also important to explore the possible speeds that a bobsled could achieve on the track. A simple but reasonably realistic model for how the speed V changes in time t is3 Mr : Mg Sin(a) i ,an 7 l,OCDAVZ. dt 2 This ODE represents Newton's law: the left-hand-side is the mass M of the bobsled times its acceleration and the right-hand-side is the sum of all the forces acting on it. The rst term on the right is the force of gravity along the s10pe with g the acceleration due to gravity and o: the angle corresponding to the slope. The second term is the frictional force given by a coefcient of friction p: times the normal force F\". The third term is air drag on the bobsled where p is the density of air, CD is a drag constant and A is the effective cross-sectional area of the bobsled. Realistic values for the parameters are M = 600 kg, 9 = lOms'l, ,u = 0.01, p = 1kgm'3, CD = 0.3, A : 0.4 m2. This model is valid for curved tracks with varying slopes, however it cannot be solved by hand in this case. Therefore for the assignment we will assume the track is straight with a constant slope. In this case Fn : Mg c0s(cr) and we will assume a: m 6.30 so that gsin(a:) 7 ,ug cos(oz) : 1. With these parameter values, the ODE becomes dvi 742 dt 1 10 V. 1. Solve this ODE to nd V(t) subject to the initial condition V(0) = 0. [3 marks] 2. What is the speed at long times if the ramp is arbitrarily long? [1 mark] 3. The distance D(t) that the bobsled has travelled along the track from the start satises the ODE Integrate this equation to nd D(t). [2 marks] 4. Plot the speed against position for a track of length 1500 H1. Again be sure to include axis labels and units on your plot. Hint: time can be thought of as a parameter for this plot. [1 mark]