Math 326 Jan 27 Model

Math 326, Jan 27 - Projectile Con Air Resistance

Today we build a model of projectile motion with air resistance, having angle and initial velocity as inputs. First, we model projectile motion without air resistance as in 3.1. We begin with an initial velocity of 30 m/s and an angle of 40 degrees. Compute the initial velocity in x (vx) and y (vy) using trig functions. Each timestep (start with timestep 0.1 seconds), update vx and vy based on acceleration ax and ay (ax will be 0, ay will be -9.81 due to gravity). Update x and y based on vx and vy. Graph the projectile path using x and y for your axes.

Now let's introduce air resistance into the model. Before, acceleration was only from acceleration due to gravity. Now drag will provide additional acceleration. For this model, we will be assuming the acceleration (or deceleration in this case) is proportional to the square of the velocity. For the proportionality constant, we use c=g/V², where g is acceleration due to gravity and V is terminal velocity.

The acceleration due to drag is a vector of length c*|v|², in the opposite direction of velocity. We use r to denote the air resistance acceleration vector.

r = -c|v|v = -(g/V²)(v_x²+v_y²)^1/2(v_x,v_y) = (-(g/V²)(v_x²+v_y²)^1/2v_x, -(g/V²)(v_x²+v_y²)^1/2v_y)

Start with V = 85 m/s, which is approximately the terminal velocity for a human who has fallen (jumped?) from a plane. Incorporate this additional acceleration into the projectile model. Include a graph of the motion and scroll bars for angle and initial velocity.

Additional Questions

Going for Distance: Determine the angle which gives the best distance. Check to see how your answer varies with initial velocity.
A Blustery Day: Introduce wind into the model. Let the wind blow in the x-direction, positive or negative. Wind will not affect the position or velocity computations (x, y, vx and vy), but it will affect air resistance computations. If the object is moving with the wind, there will be less air resistance; against the wind, there will be more.

When computing the air resistance factors r_x and r_y (as well as |v| which is only used calculating air resistance), we should use (v_x - wind) instead of v_x, which will give the velocity relative to the air. Have a scroll bar for wind, which allows for positive and negative wind (the scroll bar min can't be set negative, so you need to compensate for this).
"That throwing stick stunt of yours has boomeranged on us!": Place a high jump bar (that is, a dot) at 30 meters high, 20 meters away. Playing with the scroll bars, find an angle, initial velocity and wind speed for which the projectile will go around the bar then return to land at the starting point.
The Wind At Your Back: On the same graph, plot the path the projectile would take discounting friction. Adjusting wind speed until the graphs are reasonably matched. Could we have predicted the wind speed which would make the graphs similar?