Projectile Motion Problem Solving

Once we have established the equations of motion which describe a projectile motion trajectory, there are a number of problems we are able to solve using this. Recall that the equations of motion for a projectile object are:

\begin{aligned} \mathbf{x}(t) &= \begin{bmatrix} x_0 \\ y_0 \end{bmatrix} + \begin{bmatrix} v_{x,0} \\ v_{y,0} \end{bmatrix}t +\frac12\begin{bmatrix} 0 \\ -g \end{bmatrix}t^2 \\ \mathbf{v}(t) &= \begin{bmatrix} v_{x,0} \\ v_{y,0} \end{bmatrix} + \begin{bmatrix} 0 \\ -g \end{bmatrix}t \\ \end{aligned}

The first thing we will often do is use additional information to determine the values of the parameters in this equation. For example, we might be given the velocity and position vectors at a single point in time, or at two different points in time. Once we have this information, we would then solve the equations above by substituting in values for $t$ and $\mathbf{x}$ or $\mathbf{v}$ . Usually the value of $g$ will be given to us, but sometimes we might need to use another observation of position or velocity to determine this value.

Once we have the equations of motion, we can answer some interesting questions about the projectile. For example, we could calculate:

The time of flight for the projectile,
The range of flight,
The maximum height achieved,
The maximum speed achieved,
and many others!

In some problems, we may be given an external parameter that can be changed (such as a launch angle) and we might need to find the value of this parameter that maximises or minimises one of the above stats.

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