Math 331

Homework #6

Instructions. This is due on April 20 at the beginning of class. It is worth 20 points. Late papers lose 3 points per day late. You must work by yourself, but feel free to consult with me. Explain what you did and why you did it: good explanations are essential! Your paper should have an introduction, a main body and a conclusion. Start early, ask questions and do well.

Introduction. In this assignment, we return to the motion of objects and set up realistic equations for three-dimensional motion. We can’t solve the equations, but the TI-89 can give us accurate graphs. Assume that positive x is “to the right,” positive y is “straight ahead” and positive z is “up” and call the components of velocity v_x, v_y and v_z. The three forces are gravity, air drag and the Magnus force from the spinning of the ball. The Magnus force depends on the vector s = <s₁,s₂,s₃> which is along the spin axis. Broken down by components, the equations of motion are

Problems. For baseball, reasonable values of the drag and spin coefficients are 0.0025 and 0.005, respectively. Using the technique of section 7.1 (for each component), write out the six equations giving this as a system of first order differential equations.

Suppose a baseball is hit straight ahead from a height of 3 feet with initial speed 160 ft/s at angle 35° with no spin. Explain why the x-component can be ignored in this problem. Use the TI-89 to graph the solution (sketch a copy of the graph you get) with y as the horizontal variable and z as the vertical variable. Estimate the maximum height and range of the baseball and describe how you get your estimate. (Write down details!) Compare the amount of time the ball spends going up and the time it spends coming down.

Next, suppose a baseball is hit straight ahead from a height of 3 feet with initial speed 160 ft/s at angle 35° with spin vector <35,0,0>. This is the same as the previous situation except we now have backspin (3500 rpm). Graph the solution, describe the shape of the graph (is it symmetric?) and estimate the maximum height and range of the baseball. Explain why height and range are larger than the values from the no-spin hit.

Now, suppose a baseball is hit straight ahead from a height of 3 feet with initial speed 160 ft/s at angle 35° with spin vector <33,0,12>. Graph the solution and estimate the maximum height and range of the baseball. Using x as the horizontal variable and z as the vertical variable, determine how far the ball curves due to the sidespin. Again, describe how you get this estimate.