The question is: How can we describe the motion of an object thrown in the air mathematically? The easy way to solve that kind of problem is to separate the movement into its two components: the horizontal movement (x-axis) and the vertical movement (y-axis). Since the horizontal component is not subjected to gravity, we can describe its movement as a uniform rectilinear motion, i.e. a constant-speed motion. It is therefore simply:
$$v_x = \frac{\Delta x}{\Delta t}$$
Here, \(v_x\) is the constant-speed, and \(\Delta x\) and \(\Delta t\) are the variation in length and time.
As you may expect, the vertical component is fully subjected to gravity; we can describe its motion as a free fall. The motion is represented by the equations of uniformly accelerated movement. For example, we can use this expression to find the y-position as a function of time y(t):
$$y(t) = y_0 + v_0y t + \frac{1}{2} a t^2$$
Here, y_0 is the initial y-position, v_0y is the initial y-speed and a is the gravitational acceleration (on Earth a = g = -9.81 m/\(s^2\)). Note that a = g only because we are in a free fall situation.
To better visualize what is going on during a free-fall, watch the follow simulation:
Note that the “gravity” vector is always oriented downward, but the speed-vector changes direction after reaching its maximal height.
In short, it is the combination of these two components that produces the parabolic (arc-like) trajectory. What connects them: the time-variable (or otherwise the physics would not really work…).
Let’s explore the variation of the different parameters; here, v_0y, a and the initial angle.
Variation of the initial angle: