The Projectile Simulator embedded in this page allows one to determine the escape velocities for various bodies by observing the behavior of projectile shot vertically from their surfaces. There are many simplifications used in this simulator such as ignoring atmospheres, rotation, and other gravitational influences and the bodies are assumed to be spherical and of uniform density. The simulator will be set for the Earth in default mode, but you can simulate other bodies by changing the Mass and Radius. There is also an animation rate setting allowing you to control how time passage in the simulator maps to actual time passage.
Use the Projectile Motion Simulator to experiment with firing a high-powered rifle bullet vertically from the surface of the Earth. Would you expect such a bullet to escape from the Earth? Run the simulator for a muzzle velocity of 1400 m/s. Note how the velocity of the bullet decreases due to the gravitational influence of the Earth until it reaches its peak height. Then the velocity becomes negative and increases in magnitude as the projectile returns to Earth.
Now change the projectile velocity to 15 km/s and fire again (with a high animation rate). Note that after a day passes the velocity really doesn’t decrease any more – thus the projectile was launched with a speed greater than the escape velocity. Since a projectile fired with exactly the escape velocity will come to a velocity of zero only after an infinite amount of time, it isn’t practical to try and precisely determine these values with this simulator. However, we can estimate their values by noting the velocity range in between 1) where the particle clearly returns to Earth and 2) where the particle’s velocity clearly does not decrease with altitude and the particle clearly escapes.