The answer isn't 3

First, determine the total distance traveled: we then need to know when the journey ends. It's easy to find out that the $x$ axis is hit for a second time when $t= 4$ .

Total distance $= \displaystyle \int_{0}^{4} \left| v(t) \right| \, dt$

$v(t) = p'(t) = -2t + 4$

Due to the definition of absolute value, we can write

Total distance $= \displaystyle \int_{0}^{2} ( -2t + 4 ) \, dt + \int_{2}^{4} (2t-4) \, dt$ which comes out to $8$ .

$75$ % of $8 = 6$ .

Then, we need to solve $6= \displaystyle \int_{0}^{x} \left| -2t + 4 \right| \, dt$ , which is rewritten as $6 = \displaystyle \int_{0}^{2} (-2t + 4) \, dt + \int_{2}^{x} (2t - 4 ) \, dt$ . This gives $6 = 4 + t^2 - 4t - 4 + 8$ , and solving for $t$ , we get $t = 2 \pm \sqrt{2}$ . We reject $2 - \sqrt{2}$ because time doesn't proceed backwards--in a time perspective, time doesn't proceed from $2$ to $2 - \sqrt{2}$ since the latter is less than the former.

Finally, what's required is $\left \lfloor 1000 ( 2 + \sqrt{2} ) \right \rfloor$ , which comes out to $\boxed{3414}$ .