![]() Please note that this is a command-line appplication, it does not have any user interface at this point. The upcoming release will have a compiled binary with all required DLLs for the user to run straigh away from their Windows machine. As far as complementary libraries go, the program uses the cmath, string, iostream, fstream and oimanip libraries. The source code can be compiled using Microsoft's Visual Studio application. Any other forces (magneric, etc.) are ingored in the ODEs definition. The only 2 forces considered on the body are its weight and the drag. Projectile motions are limited in the XY plane, with +y being orthogonal to the ground and aiming away from it. It is also assumed that the drag coefficient equals with 0.5. ![]() It is assumed that the drag acting on the body is fully defined through the quadratic drag formula. Please refer to that header file if you want to increase your number of steps or run the simulation in another medium (water for example) Assumptions in Physics Simulation parameters together with global contants are defined in the constants.h header file. Step sizes are kept constant (no adaptive step-size routine). The 2nd order differential equations (ODEs) defined through the free body diagram are solved using the RK4 (Runge-Kutta 4th Order) numerical scheme. ![]() ProjectileMotionQuadraticDrag Brief Introduction & Theoryĭefines a projectile solver (well-known problem in Newtonian physics) that includes quadratic drag in C++. annotate ( 'Starting point', xy = ( 0, 0 ), xytext = ( 50, 100 ), arrowprops = dict ( arrowstyle = '->' ), fontsize = 14 ) plt. title ( 'Projectile Motion', fontsize = 16 ) plt. plot ( 0, 0, 'ko', markersize = 10 ) plt. sin ( theta ) z0 = sol = odeint ( model, z0, t, args = ( params ,)) x, y, vx, vy = sol, sol, sol, sol y = y x = x vx = vx vy = vy return x, y plot () x1, y1 = projectile ( angle1 ) x2, y2 = projectile ( angle2 ) plt. radians ( angle ) params = x0, y0, vx0, vy0 = 0, 0, v * np. array () return dz_dt def projectile ( angle ): theta = np. Import numpy as np import matplotlib.pyplot as plt from scipy.integrate import odeint from scipy.optimize import bisect def model ( z, t, params ): x, y, vx, vy = z dx_dt = vx dy_dt = vy dvx_dt = - k * vx dvy_dt = - g - k * vy dz_dt = np. sin ( theta ) z0 = sol = odeint ( model, z0, t, args = ( params ,)) x, y, vx, vy = sol, sol, sol, sol y = y x = x vx = vx vy = vy xground = x + y * ( x - x ) / ( y - y ) diff = xground - target return diff radians ( theta ) params = x0, y0, vx0, vy0 = 0, 0, v * np. vectorize def diff ( theta ): theta = np. zeros () def model ( z, t, params ): x, y, vx, vy = z dx_dt = vx dy_dt = vy dvx_dt = - k * vx dvy_dt = - g - k * vy dz_dt = np. arange ( t_init, t_final, step_size ) z = np. Import numpy as np import matplotlib.pyplot as plt from scipy.integrate import odeint from scipy.optimize import bisect #basic parametrs k = 0.01 g = 9.8 v = 100 target = 500 t_init, t_final, step_size = 0, 20, 0.001 t = np. The equation of motion -įor numerically solving the differential equations, let us split them into two first order differential equations. The projectile motion is influenced by force of gravity and drag force due to air resistance. What should be the angle of fire so that cannonball hits the target accurately? Consider, the air resistance to be approximately 0.01. Suppose, the target is at a distance 100 kilometer from the cannon. Consider a cannonball fired towards a target with a certain velocity, say $100 m/s$. ![]() In this section, I am going to simulate another kind of projectile motion cosidering effect of air resistance. In my previous post, I discussed how to simulate vertical motion of the free fall of a projectile without considering air resistance.
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