You will need to modify this code to match the requirements of your specific project, especially the initial conditions. The equations of motion are developed below and the appropriate Matlab code shown. Once this is done the “ODE45” routine can be used to solve for the x and y positions and the x and y velocities as functions of time.
Therefore the two 2nd order equations must be converted into a system of four first order equations. The function only permits a single integration, and not the double integration that is needed. The two nonlinear 2nd order coupled differential equations of motion for the x- and ydirections can be solved numerically in Matlab using the Runge-Kutta “ODE45” function. For a ping pong ball travelling at the speeds achieved in this project a drag coefficient of 0.47 ± 0.03 at 95% confidence is appropriate. The drag coefficient depends on a number of factors that will be addressed in your fluid dynamics course. Where 𝐴𝐴𝑓𝑓 is the frontal (silhouette) area of the projectile, 𝜌𝜌𝑎𝑎𝑎𝑎𝑎𝑎 is the density of air, and 𝐶𝐶𝐷𝐷 is a dimensionless drag coefficient. Recalling that 𝑥𝑥̇ (read as x-dot) is shorthand for and 𝑥𝑥̈ (x-double-dot) is shorthand for 2, 𝑑𝑑𝑑𝑑 𝑑𝑑𝑡𝑡 the two equations of motion governing the flight of the projectile are: x-direction: It will reduce to zero at the apogee, and then become negative as the projectile falls down. As the projectile flies through the air, β will start at the initial launch angle β0. This angle is measured as positive when it is above the horizontal and negative when it is below. The instantaneous angle of flight β is the same as the instantaneous direction of the velocity vector. The magnitude of the drag force changes with speed. EM375 MECHANICAL ENGINEERING EXPERIMENTATION PROJECTILE MOTION WITH AIR RESISTANCE For projectile motion where air resistance cannot be ignored, there are two forces of importance: the projectile’s weight mg which is constant and is always directed down, and the drag due to air resistance FD which is directed in the opposite direction to the instantaneous velocity.
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