Finding the difference in range of a projectile (air resistance vs. free)
We were give two equations that show the position as a function of time on the x and y axis where there is a linear drag force of 'beta*a vector v'. I have already solved for the case of no air resistance. The following are the two equations which include air resistance:
x(t) = x(t=0) + ((m*vo(x))/beta)*(1-e^((-beta/m)*t))
--> x(t=0) is 0, giving:
x(t) = [(m*vo(x))/beta]*[1-e^((-beta/m)*t)]
y(t) = [(vo(y)*(m/beta))+g*((m/beta)^2)]*[1-e^((-beta/m)*t)]-(g*(m/beta)*t)
--> y(t) would be 0, considering the full path of the trajectory results in no delta y, so setting the equation equal to 0, I was able to start solving for 't' to plug into the equation for x(t) to find a total Range.
However, I am caught on the exponential function since it includes 't'. I am not sure how to get that out of there so I can solve for 't' without including 't' in the result. This is where I am so far:
t = [(vo(y)+m)/g*beta]*[beta*(1-e^((-beta/m)*t))]
If you all could help me, I would greatly appreciate it! I'm sorry if there is a way to input equations, I am new to the forum and don't know how to do that if there is a way.
Re: Finding the difference in range of a projectile (air resistance vs. free)
You don't need to eliminate t in order to find a single function of y=f(x), but rather you have x(t) and y(t) as two parametric functions which define your projectile and you can plot it too.
It means you vary t and calculate x(t) & y(t) that give your projectile.
Trajectory would look like: