This Video Explains What Would Happen if NASA Used a Space Gun to Fire Off a Spacecraft

The experiment starts with an idea created by Sir Isaac Newton called the Newton cannonball. In this thought experiment, Newton pictured a cannon at the top of a mountain.

If he fired the cannon and there were no forces of gravitation or air resistance, the cannonball would follow a line away from earth in the direction it was fired. However, if there were gravitational forces, Newton theorized it would follow different paths depending on the initial velocity.

Too low a speed would simply send the cannon back to Earth. But if the speed was at the orbital speed at that altitude, it would circle the Earth in a fixed orbit like the moon. 

Newton's cannonball was one of the first thought experiments to think through escape velocity and ultimately how to get an object from earth into orbit. The HARP (high altitude research project) project in the 1960s was a US/Canadian project and attempt to test what it took to send a ballistic to outer space.

It was followed up with a similar SHARP project. A massive issue for both of those projects came from not having enough energy to break through the densest parts of the Earth's atmosphere and maintain enough velocity to reach orbit. 

There are plenty of other reasons we're not using rocket guns and opting instead for boosters and launching pads. Sometimes the more expensive option is scientifically the smarter choice.