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A differential equation is a mathematical expression that relates a function with its derivatives. In projectile motion, differential equations help us describe how various physical quantities change over time.
For projectiles, like a tossed ball, the motion according to Newton's laws is summarized with differential equations. They consist of initial velocity, gravity, and resistance (if any). In this context, understand that gravity affects the projectile only in the vertical direction (denoted by the vector \(-g \mathbf{j}\)). It indicates that the horizontal motion (x-direction) is uniform, meaning there's no acceleration from external forces in that direction.
By solving differential equations, specifically second-order for motion problems, we can derive expressions for the position and velocity of the projectile as functions of time. In balance with initial conditions, the differential equations yield specific formulas for the trajectory of the projectile.

When solving dynamics problems like projectile motion, we often encounter initial value problems (IVP). This is a specific type of differential equation problem where initial conditions are given. For example, a projectile's starting position and initial velocity vector.
In this problem, you start with conditions like the initial location, \(\mathbf{r}(0) = x_0 \mathbf{i} + y_0 \mathbf{j}\), and initial velocities along each axis. These stipulations allow the solving of differential equations to obtain precise motion equations. IVPs are crucial as they define the starting point for the motion and thereby influence subsequent calculations.
Solving an IVP enables us to track how the projectile moves over time. With known initial speeds and directions, combined with the gravitational pull, we can predict the complete trajectory, determining both how far and how high the projectile will go.

Vector calculus involves using vectors to study physical quantities that have magnitude and direction, a key area in projectile motion analysis. In this context, displacement, velocity, and acceleration are considered as vectors, enabling better representation and computation.
The position vector \(\mathbf{r}\) represents the projectile's location in a two-dimensional space, \(x\text{-direction}\) and \(y\text{-direction}\). Breaking down these components helps us focus separately on how the projectile travels horizontally and vertically. When analyzing motion, recognizing that vectors simplify these complex movements by providing clarity to direction makes calculation and understanding more intuitive.
Calculating the derivatives of vectors, one finds velocities and accelerations, laying out how positions change over time. This method proves especially helpful in projectile motion where different forces may act in varied directions.