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 from the ground with alpha = 45 degrees, the range (the distance at which the projectile lands back on the ground) is maximal. As for any launch angle, the trajectory is a symmetric parabola: the peak is reached exactly halfway through the flight time.

Launch angle	Shape of the trajectory
0 degrees	half-parabola if launched from a height (from the ground: no range)
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.