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*v_{o}(x))/beta)*(1-e^((-beta/m)*t))

--> x(t=0) is 0, giving:

x(t) = [(m*v_{o}(x))/beta]*[1-e^((-beta/m)*t)]

y(t) = [(v_{o}(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 = [(v_{o}(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.

Thanks everyone!

-Tina