According to the requirements in the question, PQ is perpendicular to AB or DE is perpendicular to BC. According to the question, AC = 3, AB = 5, BC = 4.

Right triangle APQ∽ right triangle ABC, so AP/AQ=AB/AC=5/3.

And CP=AQ, that is AP/CP=5/3.

That is (AP+CP)/CP=(5+3)/3.

That is, AC/CP=8/3

So CP=3AC/8=9/8.

Similarly, when DE is perpendicular to BC and PQ is perpendicular to AC, there are AP/CP=3/5 and CP= 15/8.

The point velocity is 1 unit per second, so t=9/8 seconds or 15/8 seconds.