One mechanical question, two possible solutions. Can mathematics help to predict the actual physical event?

Question

I have tried on my kid's textbook question (so to have meaningful discussion with him), but it seems to have two solutions:

My first calculation is as below:

.

Second solution:

.

Both calculated values are ok (no negative value, for example), but how can we tell which is correct without physically carrying out the experiment? (And the wording of the question seems to hint at one than the other)