The trajectory of a cannon shell

Calculate the trajectory of our cannon shell including both air drag and the reduced air density at high altitudes so that you can reproduce the results in the figure.Perform your calculation for different firing angles and determine the value of the angle that gives the maximum range.

Projectile motion

• Newton's second law in two spatial dimensions

  
  (1)

  Euler method for second-order ODE

  • write each of these second-order equations as two firest-order differential equations

    
    (2)
    • Euler method with resistance
      
      
      
      

    How to decide the point of fall?

    • the last point above the ground(n) and the first point below the ground(n+1) is given by

      
      (3)

      Effects of altitude

      • isothermal (constant temperature) ideal gas
      =p(0)e^{-mgy/k_BT})
      (4)