Why did NASA invest so much amount of time and money on developing computers for Apollo Missions? Wouldn't the Man on Moon mission have been far easier and faster if the Astronauts drove the spacecraft, in their line of sight, (like we drive cars on earth) with information from instruments such as their speed, fuel left, etc? Why weren't they in control of the flight instead of the computers; i.e. complete human control? Any specific and advantages for mission control?
The onboard computers were primarily provided for Guidance, Navigation, and Control (GNC). The mere act of starting and/or stopping a rocket engine burn is simple enough for humans to do.
Figuring out when to start and stop that burn, and where to point the spacecraft during it, are not something humans are going to do unaided in an operational amount of time. That can be calculated on the ground or onboard the spacecraft and in a practical world requires computers to do the calculating. Having the ability to do it onboard protects against communication problems with the ground and gives the spacecraft a degree of autonomy.
Line of sight maneuvers are not precise enough regarding the huge distance from Earth to the Moon. The mid course corrections were impossible without the calculations done by the computer. If course corrections far from the Moon are not precise, much more fuel is needed for corrections close to the moon.
Orbital rendezvous maneuvers are much more difficult than driving a car on Earth or flying a plane in air. If you fire the thrusters to simply decrease distance to the rendezvous target, you increase it. Decreasing the orbit height would increase the angular speed, increasing the height would decrease the angular speed.
There is no speed information from a rotating tire of a car. To measure left fuel in zero gravity could not be done like a car tank.
Try to imagine driving a car on an invisible road. If you miss the road, you will not land on the Moon nor return to Earth. All you see, the Sun, the Stars, the Earth and the Moon are too far away to give you any visual information about distances and speed.
The decision to have a complicated computer that controlled many of the spacecraft functions was made in 1961, very early in the Apollo program. From NASA's history, Computers On Board The Apollo Spacecraft: The need for an on-board computer
The presence of a computer in the Apollo spacecraft was justified for several reasons. Three were given early in the program: (a) to avoid hostile jamming, (b) to prepare for later long-duration (planetary) manned missions, and (c) to prevent saturation of ground stations in the event of multiple missions in space simultaneously.
These justifications implicitly assumed that the only viable alternative to an onboard computer was a computer on Earth that would receive navigation sensor data from and issue thruster commands to the spacecraft. Moreover,
Yet none of these became a primary justification. Rather, it was the reality of physics expressed in the 1.5-second time delay in a signal path from the earth to the moon and back that provided the motivation for a computer in the lunar landing vehicle.
A three plus second lag between inputs from sensors and commands executed in response to those sensor readings is completely unacceptable from a control theory perspective. It would have made control be outside the stability and controllability envelopes.
A basic problem with using thrusters is that nothing is perfect. Thrusters are never oriented in exactly the direction they are supposed to be oriented, the thrust produced by the thruster is never exactly the magnitude it is supposed to produce, and the center of mass is never exactly where it is expected to be.
Firing a thruster to change velocity will inherently generate undesired torques. The attitude control system must compensate for these undesired torques, and the response has to be very quick. With Apollo, those attitude adjustments came from attitude thrusters.
Even in the two flight regimes where the Apollo astronauts did have manual control, final landing and rendezvous, the astronauts did so with computer assistance. The astronauts did not directly command thruster firings. The computer instead translated joystick commands into thruster firings. The astronauts could switch between modes, with several modes where the computer automatically maintained attitude while the astronauts controlled altitude and vehicle separation. The computer automatically addressed the undesired cross couplings between force and torque.
Manual rendezvous was tried first, in Gemini 4. It went badly, because orbital mechanics are very counter-intuitive.
As GPO engineer Andre Meyer later remarked, "There is a good explanation for what went wrong with rendezvous." The crew, like everyone else at MSC, "just didn't understand or reason out the orbital mechanics involved. As a result, we all got a whole lot smarter and really perfected rendezvous maneuvers, which Apollo now uses." Catching a target in orbit is a game played in a different ball park than chasing something down on Earth's essentially two-dimensional surface. Speed and motion in orbit do not conform to Earth-based habit, except at very close ranges. To catch something on the ground, one simply moves as quickly as possible in a straight line to the place where the object will be at the right time. As Gemini IV showed, that will not work in orbit. Adding speed also raises altitude, moving the spacecraft into a higher orbit than its target. The paradoxical result is that the faster moving spacecraft has actually slowed relative to the target, since its orbital period, which is a direct function of its distance from the center of gravity, has also increased. As the Gemini IV crew observed, the target seemed to gradually pull in front of and away from the spacecraft. The proper technique is for the spacecraft to reduce its speed, dropping to a lower and thus shorter orbit, which will allow it to gain on the target. At the correct moment, a burst of speed lifts the spacecraft to the target's orbit close enough to the target to eliminate virtually all relative motion between them. Now on station, the paradoxical effects vanish, and the spacecraft can approach the target directly. Gemini IV's problem was compounded by its limited fuel supply; the Spacecraft 4 tanks were only half the size of later models, and the fuel had to be conserved for the fail-safe maneuvers. When McDivitt and White broke off their futile chase, they had exhausted nearly half their load of propellants.
Gemini 5 was used to test maneuvers and understand what happened during Gemini 4, and then the US finally accomplished rendezvous during the Gemini 76 joint mission. It was flown by the computer, for most of the closure, with the astronauts taking over for the final connection.
Three hours 15 minutes into the mission, Elliot See told Schirra that radar contact should soon be possible with Gemini VII. The VI-A crew got a flickering radar signal, then a solid lock-on at 434 kilometers range. Over Carnarvon, at 3 hours 47 minutes, the aft thrusters fired for 54 seconds to add 13 meters per second to Gemini VI's speed. The result was almost a circle, measuring 270 by 274 kilometers. In slant range distance, the two spacecraft were now 319 kilometers apart and closing slowly.
Schirra and Stafford placed Gemini VI-A in the computer (or automatic) rendezvous mode at 3 hours 51 minutes into the flight. While the lower orbiting vehicle gained slowly on its target, Schirra dimmed the lights on, his side of the spacecraft to improve outside visibility. At 5 hours 4 minutes, he exclaimed, "My gosh, there is a real bright star out there. That must be Sirius." The "star" was Gemini VII, reflecting the Sun's rays from 100 kilometers away.
Gradual catchup of the target vehicle lasted until 5 hours 16 minutes; Schirra prepared to make the last rendezvous maneuvers. The two ships were now close enough to allow Spacecraft 6 to thrust directly toward Spacecraft 7. He fired the thrusters and closed on Gemini VII at a rate of better than three kilometers every minute and a half. Schirra and Stafford briefly lost sight of Gemini VII when it passed into darkness but soon picked up the target's running lights.
Schirra made two midcourse corrections spaced 12 minutes apart (at 5 hours 32 minutes and 5 hours 44 minutes). Six minutes later, at a range of 900 meters from his target, Schirra began braking his spacecraft by firing the forward thrusters. Soon he had no difficulty seeing Gemini VII. Fittingly, in the terminal stage of rendezvous, the VI-A astronauts saw the stars Castor and Pollux in the Gemini (Twin) constellation aligned with their sister ship. Then Spacecraft 7 flashed into the sunlight - almost too bright to look at. From a distance of 200 meters, it resembled a carbon arc light. Following the braking and translation maneuver, VI- A coasted until the two vehicles were 40 meters apart, with no relative motion between them. The world's first manned space rendezvous was now a fact. In Mission Control, the cheering throng of flight controllers waved small American flags, while Kraft. Gilruth, and others of the jubilant crowd lit cigars and beamed upon this best of all possible worlds. At 2:33 p.m., 15 December 1965, Gemini VI-A had rendezvoused with Gemini VII.
Source: From http://www.astronautix.com/g/gemini6.html
So it would never have been practical to do this without computers assisting. Missed rendezvous, like Gemini 4 would have cost more than the computers.