X ,y ,duration ,maxrange = traj_fr (angles, v0 )Īngles = angles / 2 / math. zeros (n ) #generate array of n elements to take range from traj_fr for i in range (n ) : #loop to run angles through traj_fr function & populate maxrange array with distance results pi / 2, n ) #generate array of n angles print 'Angles: ', angles X ,y ,duration ,distance = traj_fr (angle ,v0 ) #define variables for output of traj_fr function print 'Distance: ', distanceĪngles = np. Y = y #truncate y array return x, y, (dt *i ), x #return x, y, flight time, range of projectile I = i + 1 #increment i for next iteration X = ( ( 2 *x -x ) + (f * x ) ) / ( 1 + f ) #numerical integration to find x. I = 1 while y >= 0 : #loop continuous until y becomes <0, ie projectile hits groundį = 0.5 * gamm * (h - y ) * dt #intermediate 'function' used in calculating x & y vals below X ,y = x + vx0 * ( 2 *dt ), y +vy0 * ( 2 *dt ) #calculating 2nd elements of x & y based on init velocity X ,y = 0, 0 #initial position at t=0s, ie motion starts at (0,0) zeros ( len (time ) ) #initialise y array zeros ( len (time ) ) #initialise x array sin (angle ) *v0 #compute y components of starting velocity cos (angle ) *v0 #compute x components of starting velocity The air density is 1.225 kg/m3 1.225 k g / m 3 (standard sea-level atmosphere) and the acceleration due to gravity is g 9.81 m/s2 g 9.81 m / s 2. Trajectories of a projectile in a vacuum (blue) and subject to quadratic drag from air resistance (red). Real Projectiles Real projectiles experience air resistance Consequences: Actual X velocity is not constant nor is X acceleration Actual Y acceleration. H = 100 #height (used to compute f, below) def traj_fr (angle, v0 ) : #function that computes trajectory for some launch angle & velocity You can change the various parameters to see their effects on the object’s motion. Gamm = 0.005 #gamma (used to compute f, below) pyplot as pltĭt = 1e - 3 #integration time step (delta t)Īngle = math. Furthermore, there is always an initial time interval during which the trajectory is identical to that calculated in the absence of air resistance ( i.e., ). Example: projectile motion with air resistance in python # -*- coding: utf-8 -*- import matplotlib. It can be seen that as the air resistance strength increases ( i.e., as increases), the range of the projectile decreases.

Projectile motion with air resistance