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.

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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.