I submitted an assignment late. Should I email the professor to explain my situation? I submitted an assignment 1 day late. Should I email the professor to explain my situation? He has a pretty loose policy on deadlines but we should definitely submit the assignment before the answer is posted.

I dropped the assignment in his mailbox so if he does not check on Sunday and posts the answer keys on the same day, he might be suspicious of me submitting the assignment after the answer key is posted, which is way less acceptable.
Tell him, he may or may not accept your submission, but if he wants to accept it and needs proof that it was before the answers came out then telling him provides that.

Don't provide a 16 page opera about why it was late, an apology, a (short) reason : medical etc and close.