1. CS 282

2.  Last week we dealt with projectile physics ◦ Examined the gravity-only model ◦ Coded drag affecting projectiles  Used Runge-Kutta for approximating integration ◦ Coded wind effect  This week we will add the last effect…

4.  First of all, open up a completed version of last week’s lab ◦ If neither you nor your partner have it, you will have to code it again. Refer to last week’s instructions.  So that we have a common starting ground, set the parameters for the following… ◦ Initial position: (0.0, 0.0, 0.0) ◦ Initial Velocity: (15.0, 20.0, 1.0) ◦ Drag Coefficient: 0.05 Mass: 10.0 Radius: 1.2 ◦ Wind Velocity: (5.0, 2.0, 0.0)

5.  Compile it and run the following scenarios ◦ IMPORTANT: Limit your simulations by either time (i.e. 5 seconds) or by position (when y position is at 0)  Gravity-only model  Gravity and Drag Model  Gravity, Drag, and Wind Model  Start outputting the magnitude of the velocity to the screen for each model ◦ sqrt (v x 2 + v y 2 + v z 2 )  Capture these outputs in separate files ◦ Use > on the command-line

6.  Spin is a very common occurrence when dealing with projectile physics. ◦ A bullet leaving a gun will have spin ◦ A golf ball will also have spin when hit by the club  Spinning objects generate force

7.  First of all, a spinning object generates lift ◦ This is called the Magnus effect (also known as Robin’s effect) ◦ The direction of the force will be perpendicular to the velocity and spin direction  If the object has backspin… ◦ Then the Magnus force will be positive in the vertical direction, propelling the object upwards  On the other hand, if the object has topspin… ◦ The resulting force will be in the opposite direction, thus pushing the object down

8.  Here we can see all the forces interacting with our ball (minus the wind)  Glossary: ◦ C L (lift coefficient) ◦ p (fluid density)  Greek letter rho ◦ v (velocity magnitude) ◦ A (characteristic area) ◦ R (spin axis vector) ◦ w (rotation velocity)  Greek letter omega

9.  We need to create some variables in our class ◦ We will need the following doubles…  Spin axis components: X-axis rotation, Y-axis rotation, Z-axis rotation  Angular/Rotational velocity (omega) ◦ And some way of setting/initializing them  You can either have them be public and manually set them, or create get/set functions (better)  Create a spin_wind_drag function ◦ It will need to take delta time as a parameter ◦ HINT: We will be using Runge-Kutta

10.  For a sphere, the force F M is defined as… ◦ F M = ½ C L * p * v 2 * A  For p and A, you can just use the values calculated in the main driver function ◦ C L = (radius * rotational velocity) / v  Solve the following in terms of acceleration (remember F = ma) and add them to their corresponding acceleration components. ◦ F Mx = - (v y /v) * R y * F M ◦ F My = - (v x /v) * R z * F M ◦ F Mz = 0  Because we will rotate the sphere along the Z-axis, this component will be 0 (e.g. 0,0,1 for the spin axis vector)

11.  Hopefully, you have now added the acceleration components to each step of Runge-Kutta.  Be sure to add your k’s appropriately to the new accelerations you added.  Let’s set the spin axis to 0,0,1  Add the appropriate line of code to the driver  Compile and run

12.  What if our spin axis is on a multiple axes? ◦ Let’s get the general form of the Magnus force  F Mx = ((v z / v) * R y – (v y / v) *R z ) * F M  F My = ((v x / v) * R z – (v z / v) *R x ) * F M  F Mz = - ((v x / v) * R y – (v y / v) *R x ) * F M  Use these forces to get your acceleration now instead of the old forces.  Compile and run the with a spin axis of ◦ (0.5,0.5, 0.0)

13.  Use a spin axis of (0.75, 1.0, 0.0) ◦ Default values on the rest  If you have completed the first exercise, you already have your data for the first three models.  Collect the data for the drag_wind_spin

14.  Plot a graph showing the differences velocity magnitude vs. time of all four models. ◦ Bonus point(s) for the models not explicitly mentioned  If you wish to use a constant delta time (instead of a timer), you may do so by replacing the measure_reset call with a flat value (i.e. 0.5)