Modelling Apart from being one of the most successful tennis players of all time, Rafael Nadal is famous for the extreme amount of topspin that he uses on his shots, particularly his forehand. Topspin is generated by hitting over the top of the ball with the tennis racquet held at an angle, instead of hitting it flat (with the racquet face at 90° to the direction of swing). Topspin makes a tennis ball drop suddenly before it hits the ground, due to the Magnus effect. The ball also then kicks forward after it hits the ground, due to the effects of friction between the spinning ball and the ground. These effect can make it more difficult for an opponent to hit a good return shot. As the ball flies through the air, the translational speed of the ball slows down due to air resistance. At the same time, the rotational speed of the ball will also slow down. In this question, we will model the second of these interactions of the ball with its environment.

a. We will start with some simple assumptions, and build up to a more complex model. In part (a), we will treat the ball as a particle (no spin), and assume that air resistance is negligible. On one particular forehand, Nadal will contact the ball 0.5 m above the ground. i. Plot the different combinations of initial angle (in degrees) vs initial speed (in km/h) that will land the tennis ball on the centre of the baseline at the other end of the court, 26.5 m away. ii. Create a graph of the different trajectories that each of these shots will take. Ensure that the different trajectories in your graph will all clear the net, which is 1.1 m high and 14.5 m away.

b. The datafile "topspin.txt" contains data on the rotational velocity of the ball at various times after the ball has left the racquet after one of Nadal's shots that was captured using high-speed video. The ball is seen to slow down its rotation due to air resistance. i. Fit a simple linear model to the data. Plot the data and the fitted curve on the same axes in a new figure. Note the value of r?, which is quite high. Print a statement explaining why this is not a good model for the data. ii.

If we assume air resistance varies with rotation speed (a OC -0), then an exponential decay curve would be an appropriate curvefitting model for a relationship between w and t (if you're not certain why this is, revise ODEs from your previous mathematics). w = Ae-Bt iii. iv. Fit this model to the data, and plot the fitted curve on the same axes as the previous question. Based on this model, predict what the initial rotational speed was when the ball left the racquet. Print this to the screen. Using an appropriate root-finding technique of your choice, find the time at which the rotational speed is predicted to be 3000 RPM. Mark this value on your graph. Write an fprintf statement explaining why you chose this method. Using an appropriate numerical integration technique of your choice, calculate how many rotations the ball will undergo before hitting the ground, assuming the time-of-flight will be within the range of 0.5-2.0 seconds. Plot this relationship. v.