we will often use this import statement and then get access to all common mathematical functions this latter statement is inserted in program named ball_angle pyfrom math import horizontal position vertical position angle atan( /xprint (angle/pi)* this program runs perfectly and produces as outputas it should at firstit may seem cumbersome to have code in librariessince you have to know which library to import to get the desired functionality having everything available anytime would be convenientbut this also means that you fill up the memory of your program with lot of information that you rather would use for computations on big data python has so many libraries with so much functionality that one simply needs to import what is needed in specific program python program with vectorization and plotting we return to the problem where ball is thrown up in the air and we have formula for the vertical position of the ball say we are interested in at every millisecond for the first second of the flight this requires repeating the calculation of : gt one thousand times we will also draw graph of versus for oe drawing such graphs on computer essentially means drawing straight lines between points on the curveso we need many points to make the visual impression of smooth curve with one thousand pointsas we aim to compute herethe curve looks indeed very smooth in pythonthe calculations and the visualization of the curve may be done with the program ball_plot pyreading from numpy import linspace import matplotlib pyplot as plt linspace( * * * ** plt plot(typlt xlabel(' ( )'plt ylabel(' ( )'plt show(