Describing Projectile Motion Using Vectors

Projectile motion is the movement of objects which are in flight and subject predominantly to the force of gravity. This describes a number of situations, such as a ball in flight, a skier or cyclist performing a jump, or water in a fountain. In each of these scenarios, there are other factors at play such as drag and frictional forces, but for simplicity we will assume that gravity is the only force being applied.

We could also consider the movement of objects such as satellites and rockets to be projectile motion. In these cases the force of gravity changes throughout the object's trajectory, which makes the problem more complicated, so we will also assume that the objects under motion are close to the surface of a planet so that gravity is unchanging during their motion.

By making these assumptions, we can now create a coordinate system which is aligned with the gravitational field impacting our motion. We can define a $y$ axis, aligned with the gravitational field, which we call the vertical axis, and an $x$ axis running perpendicular to this field, which is called the horizontal axis.

By breaking the motion up into to a vertical component which is affected by gravity and a horiontal component which is not affected by gravity, we can define the equations of motion which describe these two independent components of projectile motion.

To do this, we will use two formulae from physics:

1. Newton's second law of motion $F = ma$: which tells us the relationship between the force applied to an object, the mass of the object, and the acceleration of this object.
2. The gravitational force $F_g = mg$: which tells us the relationship between the strength of the gravitational force on an object, the mass of this object, and the strength of the graviational field.

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