This question is based on the Rocket Experiment, Case Study A. In the experiment, we collected the following data for the horizontal position, X(T), and vertical position, Y(T), as functions of time, T. Table 3: Example data collected from the rocket experiment. T (s) 0 0.567 2.567 4.567 6.567 8.567 10.567 12.567 X (pixels) 0 77 391 652 885 1074 1247 1402 Y (pixels) 0 58 252 349 361 300 166 -28 We derived an ideal model that neglected air resistance. The solution to the vertical motion without air resistance is Y(T) = -19T + v sin(0o)T, (1) where g is the acceleration due to gravity (measured in pixels s-2!), vo is the initial speed, and o is the launch angle. (a) Suppose we know that the launch angle is 0o = 0.66 rad, and initial speed is vo = 200. Use MATLAB, and the command 1sqcurvefit, to find the value of the parameter g that best fits the data in Table 3. Plot the model trajectory (Equation (1)) versus the data for Y(T) from Table 3. [5] Incorporating the effects of gravity and air resistance, the modified model for the rocket motion consists of the two ordinary differential equations, m d X dT dY d72 (2a) (2b) The model parameters include the rocket mass (m), the acceleration due to gravity (g), and a positive constant associated with air resistance (c). m = -c dX dT y = =-mg-c (b) Write down the dimensions of m, X, and T. Using the model for horizontal mo- tion, Equation (2a), determine the dimension of the constant, c. [3] (c) Introduce the new dimensionless variables Y Y 8y at 2 dy dT t = = -1 T T' where y and 7 are scaling constants with the same dimension as Y and T, respectively. Show that we can write the vertical motion model, Equation (2b), in the dimensionless form dy (3) (4) Express the scaling constants y and 7 corresponding to Equation (4) in terms of the model parameters m, c, and g. [4] dt'