Final answer:
The slowest speed a spaceship needs to start with, launched from the Moon, to leave the Solar System would be higher than the escape speed of Earth, but still an extremely high velocity.
Step-by-step explanation:
The slowest speed a spaceship needs to start with, launched from the Moon, in order to leave the Solar System would depend on various factors, such as the gravitational pull of the Moon and the velocity required to escape the Sun's gravitational grip.
To escape Earth, a spacecraft must achieve escape speed, which is about 11 kilometers per second (about 25,000 miles per hour). However, even if a spacecraft achieved a higher speed, it would still take thousands of years to reach the distance to the nearest star.
Therefore, the slowest speed a spaceship needs to start with, launched from the Moon, in order to leave the Solar System is likely to be higher than the escape speed of Earth, but it would still be an extremely high velocity.