Suppose you have a hypothetical spacecraft launched from Earth.
When it escapes from Earth's gravity it will still be orbiting the sun at roughly the same velocity as the earth, approximately 30km/s relative to the center of the sun. Say the spacecraft then thrusts in the opposite direction, until its velocity relative to the center of the sun is reduced to 0 while keeping its other positional attributes as close to unchanged as possible. Now the hypothetical spacecraft is (for some brief instant in time) essentially stationary relative to the sun.
My question is, what happens next? I'd expect that the spacecraft would immediately begin to fall directly into the sun. Is that the case, and if so, how long would it take to get there and what would the acceleration profile be like?
And if one were to deliberately send a probe into the sun, is there any practical merit to doing it that way as opposed to simply thrusting directly towards the sun while retaining the lateral velocity imparted by the earth?