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projectile is a body that is projected by an external force and continues in motion by its own inertia. Some common examples of projectiles are cannonballs, baseballs and rocks that are either thrown or launched into the air. In the simplest examples, once an object has been launched or thrown, the only force the object experiences is due to gravity. When anaylzing these examples, the path of the projectile can be determined exactly using the
constant acceleration equations of motion:
\[\begin{aligned}d &= vt + \dfrac{1}{2}at^2 \\ v &= at \\ a &= \text{constant} \end{aligned}\]
While the constant acceleration equations can be used to describe simple projectile motion, they can not be used to describe projectiles that experience drag or spin because projectiles that experience either spin or drag do not have a constant acceleration. To calculate the path of a projectile that experiences drag or spin, the differential equations of motion must be used:
\[\begin{aligned}v &= \dfrac{dx}{dt} \Rightarrow dx = v \cdot dt\\ a &= \dfrac{dv}{dt} \Rightarrow dv = a \cdot dt\end{aligned}\]
The advantage to using the differential equations of motion over the constant acceleration equations of motion is that the differential equations can be used to take drag and spin into account; the disadvantage to using the differential equations of motion is that it is difficult to determine the path of a projectile exactly. To resolve this issue, an approximation of the path of a projectile can be determined by using the Euler method for solving first order differential equations. Using the Euler method, the above equations can be approximated by the following:
\[\begin{aligned}dx &= v \cdot dt \approx \\ \triangle x &= v \cdot \triangle t \Rightarrow \\ x_f-x_i &= v \cdot \triangle t \Rightarrow \\ x_f &= x_i + v \cdot \triangle t \end{aligned} \] \[ \begin{aligned} dv &= a \cdot dt \approx \\ \triangle v &= a \cdot \triangle t \Rightarrow \\ v_f-v_i &= a \cdot \triangle t \Rightarrow \\ v_f &= v_i + a \cdot \triangle t \end{aligned}\]
Lastly, to account for drag and spin, a non-constant acceleration is used. Drag and spin are dependent on the velocity of the projectile: the faster the projectile, the more drag it experiences and the larger the affects of spin.
\[\begin{aligned} \dfrac{da}{dt} &= -(drag+spin) \cdot v^2 \Rightarrow \\ da &= -(drag+spin) \cdot v^2 \cdot dt \approx \\ \triangle a &= -(drag + spin) \cdot v^2 \cdot \triangle t \Rightarrow \\ a_f &= a_i -(drag + spin) \cdot v^2 \cdot \triangle t\end{aligned} \]
We now have a set of equations that can calculate final position, velocity and acceleration based on a previous position, velocity and acceleration. Using these equations, we can approximate the current motion of a projectile based on its previous motion. By repeatedly applying these equations, we can approximate the entire path of the projectile.
\[ \begin{aligned} x_f &= x_i + v \cdot \triangle t \\ v_f &= v_i + a \cdot \triangle t \\ a_f &= a_i -(drag + spin) \cdot v^2 \cdot \triangle t \end{aligned} \]