Assignment Task:

This project will require you to use your programming expertise to solve Newton's laws of motion. There are

some systems for which the equation of motion $($Newton$\text{'}$s second law$)$ cannot be solved analytically. We will

need numerical methods for solving the equation of motion. The project involves setting up the equations for

the given system and implementing an algorithm for solving the equations of motion.

In this project, you will need to simulate the motion of a projectile $($thrown with a velocity $v_0$ at an angle $$

horizontally$)$ moving under the influence of a constant gravitational force $($in the downward direction$)$ and air

resistance. The air resistance opposes the motion $($acts opposite to the direction of the instantaneous

velocity$)$ and is proportional to the square of the velocity $($proportionality constant is b$).$

Write down Newton's second law for the $x$ and $y$ directions and then solve them by Euler's method. In Euler's

method, use the value of acceleration $($which is calculated from Newton's second law$)$ to change the velocity

using the equation $v(i+1)=v(i)+a(i)t.$ Here, $v(i)$ is the velocity at the beginning of the time

interval $t$ and $v(i+1)$ is the velocity at the end. Use the value of $v$ to change the position

$x(i+1)=x(i)+v(i)t.$

Put both these equations along with the expression for the acceleration inside a FOR loop. Remember that

you will have two sets of equations $($one set along $x$ and the other along $y).$ Store the values of $x$ and $y$ you

obtain in arrays $($or write it on to a file$).$ Use plotting software such as Microsoft Excel to draw a graph of $x$ vs$.y$

$.$ You can also do the entire project in Excel. If you use Excel, you will not have FOR loops. Instead, you will

have columns for $a_x,a_y,v_x,v_y,x,$ and $y.$