# Bound and Unbound Trajectories

An object 'm' in the gravitational field of another object

# Conic Sections

In physics, the paths that can be taken by an object under gravitational attraction are called the conic sections. In mathematics, these curves are obtained by taking a slice from a cone at different angles. These shapes are the circle, the ellipse, the parabola and the hyperbola. The circle and ellipse are **bound orbits** (like planets around the sun), and the parabola and hyperbola are **unbound** (like a deflecting rocketship on a slingshot orbit).

**Figure 1:** Illustration of the four different conic sections.

# Calculating Bound and Unbound Trajectories

To determine whether an object with mass 'm' will follow a bound or unbound trajectory, it is useful to apply conservation of energy and calculate the kinetic and gravitational potential energy of the object in the gravitational field of mass

E_{kinetic} > E_{potential} : Object 'm' has enough energy to escape the gravitational pull of object

E_{kinetic} < E_{potential} : The velocity of object 'm' is too small to escape the gravitational attraction of object 'M' and is stuck in freefall around object

The kinetic energy _{kinetic} _{potential}