In this case, "try" and "try out" both mean "to test/examine".
You have to look at what comes after the "try":
- try + noun = test: Shall we try the new restaurant?
- try out + noun = test: I want to try out the new Mercedes.
- try out + for = attempt: I'm going to try out for the football team.
- try + on = test clothing/accessories: Try on this perfume and see if you like it.
- try + infinitive = attempt: Let's try to get seats close to the stage.
Check your question to see how I edited your example questions.
to ponder
means to, over a period of time, think about the consequences of two or more choices in a decision. If there is only one thing you are pondering, you are pondering whether or not to do that thing.
to mull over
is the same as ponder - but you can mull over collective nouns, meaning you are trying to decide which one of the group. "I mulled over the candy, and found a good piece."
to contemplate
means to think about the consequences of something carefully, thoroughly, and for a long time. Typically when you contemplate something, you are deciding whether or not to do it, not picking among an array of choices. It has a serious connotation, likely due to the common phrase contemplate suicide.
to cogitate
I don't hear this word a lot. According to Google, it can be used intransitively, so you don't have to be cogitating about anything in particular. The other words here generally require a "target" to make sense.
to deliberate [on]
means the same as contemplate. When I hear this word, I think of court proceedings or meetings ("the jury will deliberate... etc.").
to ruminate
I don't hear this in speech very much. In my opinion it's a "less serious" version of contemplate.
to dwell on
means to think about something over and over. It could be the consequences of a decision not yet made, a decision that just has been made, or it could be going over past events in your mind. Dwell on can have a negative connotation, possibly that what you are dwelling on is bothering you or haunting you.
So, I would answer like this:
("depth 10k worths" doesn't make sense to me, so the second is a guess)
I have been dwelling on my son (all day) since he left home for college.
I have been mulling over my depth 10k worths because I will have to pay it by next week but I don't have enough money. So I have to find a way.
I have been dwelling on my new school project. I don't know where to start.
The government deliberated how to stop protest on raising student fees.
Best Answer
Any one of the suggested phrases above sound good to me (British), including your original offerings. They don't come much simpler than that, except maybe:
Since your family undoubtedly know what you have just finished (or where you're leaving from).
I'd simply like to add a few more alternatives to the list.