And for reference, the earth escape velocity from the surface is 11.2 km/s or 25,000 mph, not 7,000 mph.

To escape the solar system from the earth surface, the minimum speed is 16.6 km/s, or 37,100 mph. But this assumes that you launch in the correct direction to take the most advantage of the Earth’s 30 km/s. If you launch in the most disadvantageous direction, you can add another 60 km/s to escape.