I am given an initial x and y position and initial velocity and I was asked to graph the trajectory in 1 second intervals.
This is what I have so far:
If $x_0 = 1, v_{0x} = 70, y_0 = 0, v_{0y} = 80, a_x = 0, a_y = -9.8$, and time will be 0,1,2,3… and so on.
Using these equations on every second you can find the plot will be a bell shaped with the highest point being ~ 325 m at about 600 seconds:
$$ x = x0 + (v_{0x})t + 1/2((a_x)t^2) $$
$$ y = y0 + (v_{0y})t + 1/2((a_y)t^2) $$
Usually in physics, we are taught in perfect condition with no air resistance. But what if there was air resistance?
How would it affect this problem? How can i add it to my calculations and see what the difference is?
Best Answer
With drag force $- \alpha \left|{\dot{\bf r}}\right| {\dot{\bf r}}$ and gravitational force $-mg {\bf {\hat{y}}}$, the equations of motion are (see my answer to this question) $$ \begin{align} {\ddot{x}} &= - \beta {\dot{x}} \sqrt{{\dot x}^2+{\dot y}^2} \\ {\ddot{y}} &= - g - \beta {\dot{y}} \sqrt{{\dot x}^2+{\dot y}^2} \end{align} $$ where $\beta = \alpha / m$, $\alpha = \rho C_d A / 2$, $\rho$ is the air density, $C_d$ is the drag coefficient, and $A$ is the cross-sectional area of the projectile.
I've never seen the above equations solved analytically. Here's some Python code that plots the solution with and without air resistance. To run, copy into a text file myFile.py and run
python myFile.py
from the command line.