Scientific Pursuits

 is a Fallout 3 quest. It is also an Xbox 360 achievement.

Gain entrance to Rivet City
Stock up on as much supplies as you can, such as ammo and stimpaks before you set out on this quest. You have a long journey ahead of you. Exit the GNR building and proceed south along the eastern bank of the polluted Potomac river. You'll encounter some raiders and mutated creatures along the way. If you think you can take these guys in a fight, go ahead and do it. Otherwise, stick to the plan of traveling south, traveling past the Jefferson Memorial, all the way to the large beached aircraft carrier known as Rivet City. You will run into Super Mutants and Centaurs along the way.

Once you reach this area, ascend the large platform via the stairs and interact with the intercom to have the bridge from the carrier extend out to your location. Walk across the bridge and speak to Harkness. Tell him that you're here to see Doctor Li, and he'll let you in.

Once you're inside, follow the signs labeled Science Lab. Once there, walk down the stairs to the lower level of the lab and speak to Doctor Madison Li.

While here, there are side-quests. The Replicated Man quest is primarily done here. The Wasteland Survival Guide quest has a section on the history of Rivet City.

Talk to Doctor Li about Dad and Project Purity
Doctor Madison Li might be a little standoffish at first, but once she realizes who your father is, she warms up, and lets you know all about Project Purity, and where you dad ran off to. If you want more dirt on the plot details, ask her about Project Purity and all the stuff your dad was working on. For even more dirt, break into her room (outside of the science lab) and retrieve 3 tapes recorded by James.

If you want to bypass most of this plot stuff, just ask her where your dad ran off to not too long ago, and she'll tell you he's at the old lab at the Jefferson Memorial. You can fast travel there since you discovered it while walking south along the Potomac-- but don't fast travel there yet. Instead, follow the signs to the marketplace inside Rivet City to restock your ammo and stimpak supply. If you're extra nice to Doctor Li, she might even give you some supplies for your journey. Once you're all set, fast travel to the Jefferson Memorial.

Look for Dad in Project Purity's Control Room
Enter the memorial through the metal door labeled gift shop. Inside, you'll fight more super mutants and even centaurs. Use the columns in the area for cover from the super mutants and centaurs. Make your way to the rotunda by following the signs.

Like Rivet City, follow the signs to make your way to the Rotunda.

Once you're in the rotunda, walk up to the auxiliary filtration unit (it's labeled as such) and pick up the log tapes resting here. One of these tapes is journal entry #10. And even though the quest says you need to find his holotapes, you need to find only one in order to continue to the next part of the quest. This would be holotape journal entry 10, located in the place where you'll discover that your father has gone off to find another Vault 112.

Tossing mines out in front of the feet of charging super mutants will slow them down -- or kill 'em.

This Vault location will be highlighted on your Pip-Boy map. Fast travel to the nearest location to this waypoint to save you time, ammo, and stimpaks. You could walk it, but…it's a far walk, so we don’t recommend that route. You could even fast travel back to Megaton to buy up some supplies as well, because it's still a bit of a journey to Vault 112, which is filled with raiders and the like.

You can find Journal Entry #10 inside the Rotuna.

Gain access to Vault 112
Once you're near the waypoint, you'll discover the entrance to Vault 112 is disguised as a gas station. Enter it, and kill the mole rats, and then rest on the nearby bed to recoup some of your health, rather than using a stimpak.

In the corner of this area is a control switch. Pull this switch to open up the secret entrance into Vault 112. Walk down the stairs, and up to the large Vault door. Activate the control switch to have the Vault open up for you, allowing you to walk inside.

Talk to the robobrain to pick up a Vaul-Tec suit.

You'll be greeted by a robobrain, and be given a Vault 112 jumpsuit -- wear it. Continue down the stairs and through the rest of the vault until you come across the clinic (follow the signs). You'll see a large room full of sleeper pods.

What the hell has been going on here...

Drop down to the floor below. Find the empty "tranquility lounger" that is open. Step inside, activate it, and you'll be transported off to Tranquility Lane, ending this chapter. This begins the Picking up the Trail quest.

After you finish the quest "Scientific Pursuits" go to Elder Owyn Lyons and talk, after you talk to the Elder you have to go to talk a Sciencist named Scribe Rothchild. He will tell you to go down to the A and have to look for the Vault-Tec computer. It took me a long time to find which Vault had the G.E.C.K. The Vault is Vault 87. Go talk to Rothchild and he will tell you what to do from there.

Speculation
It may be possible to finish this quest without ever entering tranquillity lane. Within the Tranquillity Lounger room can be found a locked side room containing a large quantity of supplies aswell as the password for the Overseers office in which can be found Stanislaus Braun's tranquillity lounger. The door to the supply room can be opened by hacking the hard terminal directly adjacent to its door. This approach has, however, not yet been confirmed.

EDIT: I tried this approach prior to entering Tranquility Lane, and there doesn't seem to be anything in Braun's room which could enable you to skip Tranquility Lane.