An interesting hypothetical question. Please do not attempt this experiment unless you are a professional.
The Earth orbits the Sun at a speed of about 20 km/sec or so such that its velocity vector is everywhere tangential to its orbit from instant to instant. If the velocity vanished, then for an instant, the Earth would have no net radial or tangential velocity with respect to the Sun. Within a few moments, however, the Sun's gravitational field which causes a steady radial acceleration of all bodies in its vicinity, would begin to accelerate the Earth on a radial path towards the center of the Sun. There would be no tangential velocity component because the gravitational field is purely radial. If we take the mean acceleration during the trip as the acceleration at 1/2 the current orbital distance, then this amounts to about
G Msun
Acceleration = -----------
2
R
and for R = 46 million miles and M = 2 x 10^33 grams, you get ( after
conversion to standard units) an acceleration of 2.8 centimeters/sec/sec.
To travel 46 million miles ( 7 x 10^12 centimeters) will take
d = 1/2 aT^2 or about 0.9 months, so the full trip to the Sun will take a
little under 2 months. It's final velocity after a 2 month voyage would be
about 2.8 cm/sec/sec x 5.2 x 10^6 sec = 144 kilometers/sec.
More detailed calculations are, of course, possible, but the order of magnitude of the above estimate is probably correct.