Scientific Pursuits

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

Gain entrance to Rivet City
It is highly likely that you have already visited Rivet City, likely as part of The Wasteland Survival Guide or in the course of visiting the southern Metro Stations to reach areas like Museum of American History or the National Archives. In this case, reaching Dr. Li is trivial.

Otherwise, press south from the Galaxy News Radio building moving south and following the eastern bank of the Potomac River towards the Jefferson Memorial. Be prepared to fight a large number of Talon Company mercenaries, Centaur, and potentially Super mutants as you go. Rivet City is a large aircraft carrier, and is very difficult to miss.

Once you reach this area, ascend the large platform via the stairs and interact with the intercom to have the bridge from the carrier extended. Tell the security chief Harkness that you're here to see Doctor Li, and he will let you in.

The Swimming option
Another option for getting to Rivet City at the very start of the game (useful for getting the Bobblehead - Intelligence right off the bat) is to head northeast from Vault 101 until you get to the river and swim all the way down to the aircraft carrier, ducking underwater if you come under fire -- there is one section of the river where some raiders on the DC side are fighting with some mutants on the Virginia side where you'll want to do this.

Surprisingly, this is the easiest, most survivable route. If you don't dally, you will "only" pick up ~400 rads by the time you get to Rivet City. You can then fast travel back to Vault 101, head to Megaton, and do the Radiation Sickness quest portion of Moira's The Wasteland Survival Guide quest to get all of your accumulated radiation removed for free.

Once you're inside, follow the signs labeled Science Lab. Dr. Madison Li is on the lower level of the room.

Talk to Doctor Li about Dad and Project Purity
Dr. Li knows who your father is and offers details on Project Purity, then tells you where your father James ran off to: the Jefferson Memorial. If you've been to this location and visited the underground tunnel, the game will allow you to tell her that he isn't there; she will instead insist that you go back as his notes were left there.

Look for Dad in Project Purity's Control Room
Enter the memorial through the metal door labeled Gift Shop. Be warned that there are a number of Super Mutants inside. Make your way to the Rotunda by following the signs.

Once in the Rotunda, walk up to the auxiliary filtration unit near the entry way and pick up the log tapes resting here. One of the tapes, #10, will indicate that your father has left for Vault 112 already. The location will be added to your map; there are several other points of interest in the area, which makes it relatively simple to fast travel to.

Gain access to Vault 112
There is a large group of raiders outside before you get to Vault 112 so come prepared. The waypoint is revealed to be Smith Casey's Garage. Inside, the player will find several mole rats and a control switch, which allows access to a lower level leading directly to the Vault 112 door. Activate the control switch to open the Vault and step inside. A Robobrain will walk up to you, insisting that you don a Vault 112 Jumpsuit, which it will give you before asking that you please enter a Tranquility Lounger.

Continuing downstairs, you will find a number of Tranquility Loungers in a circle. One is unlocked. Opening it and sitting in the pod will end Scientific Pursuits and begin Tranquility Lane.

Rumours
It is rumoured that it is possible to skip this quest by managing to get into Braun's office without ever sitting in a Tranquility Lounger. This is not true; there is nothing in that office which will end the simulation and release James. The only way to end the quest is to enter Tranquility Lane via a Tranquility Lounger.

It is also be possible to complete the quest without gaining the achievement. Through over exploring, and failing to use the intercom for the Rivet City bridge, its possible to bypass the Rivet City section completely, as you can stumble across the journals in the Jefferson Memorial (Project Purity HQ) without ever having met Madison Li... as not all of the objectives have been completed in order this disqualifies you from receiving achievement.

Also you can just head straight to vault 112 as soon as you receive the quest and it will update the quest to that point.