The parabolic trajectory

A characteristic curve

By combining the two equations of motion x(t) and y(t) from the previous lesson, we can eliminate the time t to obtain the equation of the trajectory, that is, y as a function of x. We find an equation of the form:

y(x) = ax^2 + bx

with a and b constants that depend on v0, alpha and g. This form is that of a parabola, the same curve studied in mathematics for second-degree functions.

Why a parabola?

Starting from x(t) = v0*cos(alpha)t, we isolate t = x / (v0cos(alpha)), then substitute it into y(t). The term in t^2 becomes a term in x^2, which naturally gives a parabola opening downward (the coefficient a is negative because it contains -g).

Concrete example

For a launch with alpha = 45 degrees, the trajectory is a symmetric parabola: the range (the point where the projectile lands back on the ground) is located exactly halfway between the starting point and the peak of the trajectory, in terms of flight time.

Launch angle	Shape of the trajectory
0 degrees	very flattened parabola
45 degrees	"balanced" parabola, maximum range
90 degrees	vertical trajectory (no range)

Common pitfall

Do not confuse the trajectory (the curve of y as a function of x, a snapshot of the path traveled) with the graphs of x(t) or y(t) separately (which show the evolution over time). These are three different curves that answer different questions.