1. CS 282

2.  Review of projectile physics  Examine gravity projectile model  Examine drag forces ◦ Simulate drag on a projectile  Examine wind forces ◦ Simulate wind affecting a projectile

3.  Acceleration ◦ F = ma  Translational velocity and state ◦ dv/dt = a, ds/dt = v  Rotational acceleration ◦ torque = inertia * rotational_acceleration  Equations of motion are separated into directional components

4.  Let’s use Cartesian Coordinates for simplicity

6.  F x = 0  F y = -mg  F z = 0  v x = v x0  v y = v y0 –gt  v z = v z0  a x = 0  a y = –g  a z = 0

7.  Please sit next to your partner (if possible)  Download the framework code for today ◦ Available on the class website  Examine the projectile class ◦ Look at the gravity_model function ◦ !!!! Fill in the missing update y-position !!!!  Compile and run ◦ If necessary, WASD2X translates the camera (use at your own risk though ) ◦ ENTER toggles the simulation ON/OFF

8.  Hopefully, you have come up with something like (without cheating) this… ◦ position.y + v y *dt+ 0.5*g*dt 2  For this lab, let’s just assume there’s a golf club (or something) is hitting that projectile  So, Gravity-Only models are… ◦ Really easy to implement ◦ Feasible to analytically solve ◦ Really unrealistic looking

9.  The only force on the projectile is gravity ◦ Only acts on the vertical (or the y) direction  The motion in the three directions is independent. ◦ What happens in the y-direction, for example, does not effect the x- or z-directions  The velocity in the x- and z-directions is constant throughout the trajectory  The shape of the trajectory will always be a parabola

11.  What is drag? ◦ The resistance that air or any other type of gas exerts on a body traveling through it.  Drag directly resists velocity ◦ X, Y, and Z velocities

12.  Drag has two components ◦ Pressure drag: Caused by the differences in pressure between the front and back of the object ◦ Skin drag: As the projectile is moving through space, friction is created between it and the gas

13.  The drag coefficient, C d, is a scalar used to evaluate drag force.  The shape of an object greatly affects how much drag affects it.

14.  Examine the projectile class ◦ You will notice a function called “drag_model”  This is where you will drag will be implemented ◦ You will also notice several data members in the class that are related to drag.  Part of the drag_model function should look suspiciously familiar. ◦ Indeed, it is our favorite Runge-Kutta method!  Or at least, a fragment of it…

15.  The goal for this exercise is to finish the rest of the Runge-Kutta approximation of drag. ◦ The first step of runge-kutta is provided for you  Here are some relevant equations: ◦ F x = -F d (v x /v)Fy = -mg -F d (v y /v) ◦ F z =- F d (v z /v)  The force due to drag is ◦ F d = ½ p *v 2 *A *C d ◦ 0.5 * density * velocity 2 * cross-area * drag coeff.

16.  First, finish steps 2 through 4 of the Runge- Kutta procces ◦ Use step 1 and the equations as a reference  Be careful! Because we have more than one velocity (as opposed to just x-velocity last time), you will have k’s for each component  Don’t forget to average your k’s before you add it to the velocity, and update your position.

17.  Drag force acts in the opposite direction to the velocity. The magnitude of the drag force is proportional to the square of the velocity  Drag causes the three components of motion to become coupled (i.e x depends on y and z)  The drag force is a function of the projectile geometry and is proportional to both the frontal area and drag coefficient of the projectile

18.  The acceleration due to drag is inversely proportional to the mass of the projectile. Other things being equal, a heavier projectile will show fewer drag effects than a lighter projectile.  The drag on an object is proportional to the density of the fluid in which it is traveling.

20.  Now that we have drag, it will be relatively easier to add wind into our simulation.  The presence of wind changes the apparent velocity seen by a projectile  Tail-wind adds to the velocity of the object.  Head-wind subtracts from the velocity

21.  Luckily for us, since we’ve implemented drag already, it will be easy to add wind. ◦ You may have already noticed a function, as well as parameters, hiding in the projectile class relating to wind.  First of all, let’s keep all the work we did for drag. Go ahead and copy paste the contents of the function into the blank function “drag_wind_model”

22.  Now, all we have to do is subtract the wind’s velocity components from each section of the Runge-Kutta steps  Tail-winds will have negative velocity, thus increasing our end velocity  The reverse for head-winds

23.  Save your finished product somewhere. ◦ Perchance commit it to a repository?  You will be needing this for next week when we add on… ◦ Spin ◦ Different shaped objects ◦ Mystery?

24.  Plot the differences between the following combinations on a graph (position vs. time) ◦ Gravity only ◦ Gravity and drag ◦ Gravity and tail-wind ◦ Gravity, head-wind, and drag  Not due next week, but you will be adding more things to your simulation, so having this done will lesson your workload next week.