# CS 282

 ◦ ◦ ◦ 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…

 ◦ 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)

 ◦ 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

 ◦ ◦ 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

   ◦ ◦ 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

 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

 ◦ ◦ 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

 ◦ ◦ 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 F My = - (v = - (v = 0 y x /v) * R /v) * R y z * F M * F M F Mz 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)

 Hopefully, you have now added the acceleration components to each step of Runge-Kutta.

 Let’s set the spin axis to 0,0,1  Add the appropriate line of code to the driver  Compile and run

 ◦ What if our spin axis is on a multiple axes?

Let’s get the general form of the Magnus force    F Mx F My F Mz = ((v z = ((v x = - ((v x / v) * R y / v) * R z / v) * R y – (v y – (v z – (v y / v) *R z ) * F M / v) *R x ) * F M / 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)

 ◦ 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

 ◦ 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)