IF, and only if, you are ALREADY stationary in space, relative to the sun. In fact, you don't have to fly at all, just let gravity pull you in! You'll fall sunward in a straight line, like a brick falling off a building.

The problem is that NOTHING is stationary in space. If you took off from earth, you already share the earth's orbital velocity about the sun, about 18 1/2 miles per second. If you aim at the sun and fire your rockets, you will simply add a small component of delta-vee at right angles to your path, which will only serve to slightly flatten your near-circular orbit around the sun into an ellipse--one that still goes nowhere close to hitting the sun.

In order to "fall into the sun" you'd have to flip your ship around and fire your rockets into your direction of travel, and come up with a delta-vee of 18 1/2 miles per second, an enormous energy expenditure. (Remember, the earth's escape velocity is 7 miles per second.) This would then cancel out your forward progress entirely, and you'd fall straight into the sun! But you'd have to lift all the propellant you'd need for that maneuver off the earth's surface in the first place. We just don't have that kind of propulsion capability. In fact, even if you could miraculously put a fully fueled Saturn/Apollo multi-stage rocket into space, it still wouldn't have the suds to send a payload into the sun.

Going down gravity wells from orbit takes exactly as much energy as it takes to climb out of them into orbit. If it wasn't for atmospheric braking, we'd never be able to return to earth from LEO.

However, we ARE clever enough to exploit gravitational assists from other planets.

OH, and BTW, I remember distinctly, the first time I ever listened to the Floyd I told myself "This is the kind of music starship sailors will relax to when they're off watch." Thanks for the memories.