It's easier to use Newton's Laws of motion (SUVAT) then it is to integrate.
You are given and want to find . Thus you may use the equation solving for s.
If you use calculus then
You need to add a constant here since you integrated with respect to t: . You are told that at the time of collision (t=0) the speed is
You can use this information to find from the known values of t and v(t)
To get displacement integrate again.
I'm not entirely sure how to find c_2.