The weight flies off in a straight line because there is now no force to hold it in its circular motion. That force in the string is enough to make the weight at the end of the string circle around you. A little vector math shows that the acceleration of the weight is toward the center of the circle. The magnitude is the square of the velocity divided by circle's radius. Look at an intersting example of circular motion.
Gravity acts like a string to hold satellites in a circle.
As our rocket carries the spacecraft swiftly through the Earth's atmosphere, it is also trying to pull away from the Earth's gravitational force. The Earth's gravitational force pulls at the spacecraft like the tension in the weighted string. The Earth's gravity becomes an external force acting on the spacecraft. The direction of the spacecraft is no longer directly straight up from the Earth's surface. Let's say that the velocity of the spacecraft and the gravitational force of Earth have just the right balance.