(30 points) In baseball: a pitcher throws a baseball at a high speed um a: 40 m/s toward the strike zone at home plate which is a distance L a: 20 m away. 'We will assume, for simplicity. that when the pitcher releases the ball {at time t = U], the ball is moving exactly horizontally. Due to gravity and drag force on the ball. however: the ball will slow down and sink somewhat as it moves toward home plate. The methods you will Ilse in this problem can be easily extended to account for a small departure of the release velocity from a strictly horizontal direction1 as well as for the effects of spin on the hall. The equations we will use to model these forces are: \fBesides the pitch speed 'th and distance L to home plate previously mentioned, the variables and parameters here are as follows: We a: 150 g is the mass of the baseball. 11 and 'u are, respectively, the horizontal and vertical components of the velocity of the baseball. t is time since the release of the baseball by the pitcher. "r' a: (1.01 g/ cm is the friction constant of the baseball; it has the same meaning as previously in the class. The reason why it multiplies a more complicated expression here is that the drag force (which has magnitude 7ft!\" +1.? has to be projected onto the horizontal and vertical directions. 9 = 9.8 m/s2 is the gravitational constant giving rise to a gravitational force mg downward on the baseball. We are interested in estimating the motion of the ball between the pitcher and home plate. The equations (4) are too difcult to solve exactly. but we make the following two observations which suggest the possibility of a perturbation theory calculation: The drag force and gravitational force only slightly affect the ball's motion. c The magnitude of the frictional drag force is roughlyr comparable (but not exactly equal) to the magnitude of the gravitational force on a pitched baseball. {a} {20 points) Perform an intelligent nondimensionalization of the problem to pre- pare for a perturbation theory based on these observations. (b) {10 points) What would you take as the small parameter in the perturbation theory? Under what conditions would you expect the perturbation theory to be accurate