6.1: Projectile Motion

The simplest case of two-dimensional motion occurs when a particle experiences a force only in one direction. The prime example of this case is the motion of a projectile in Earth’s (or any other planet’s) gravitational field as locally described by Galilean gravity (Equation 2.2.2): \(\boldsymbol{F}=m \boldsymbol{g}\). Once a projectile has been fired with a certain initial velocity \(\boldsymbol{v}_{0}\), we can find its trajectory by solving the equation of motion that follows from Newton’s second law: \(m \boldsymbol{g}=m \ddot{\boldsymbol{r}}\). We can decompose \(\boldsymbol{r}\) and \(\boldsymbol{v}_{0}\) in horizontal (x) and vertical (z) components; each of them has its own one-dimensional equation of motion, which we already solved in Section 2.3. The horizontal component experiences no force and thus executes a simple linear motion with uniform velocity \(v_{0} \cos \theta_{0}\), where \(\theta_{0}=\arccos \left(\boldsymbol{v}_{0} \cdot \hat{\boldsymbol{x}}\right) / v_{0}\) is the angle with the horizontal under which the projectile was fired and \(v_{0}=\left|\boldsymbol{v}_{0}\right|\) the initial speed. Likewise, because the acceleration due to gravitation is constant, our projectile will execute a uniformly accelerated motion in the vertical direction with initial velocity \(v_{0} \sin \theta_{0}\). If the projectile’s initial position is \(\left(x_{0}, z_{0}\right)\), its motion is thus described by:

\[\boldsymbol{r}(t)=\left(\begin{array}{c}{x(t)} \\ {z(t)}\end{array}\right)=\left(\begin{array}{c}{x_{0}} \\ {z_{0}}\end{array}\right)+v_{0}\left(\begin{array}{c}{\cos \theta_{0}} \\ {\sin \theta_{0}}\end{array}\right) t-\left(\begin{array}{c}{0} \\ {g}\end{array}\right) \frac{1}{2} t^{2}\]