The general trajectory of a point mass in a constant gravitational field is a parabola.
Suppose at time 𝑡 = 0, the trajectory passes through the origin of the coordinate
system 𝑥, 𝑦 with velocity 𝒗 = 𝑣 𝒆 + 𝑣 𝒆 = 𝑣 cos 𝜃 𝒆 + 𝑣 sin 𝜃 𝒆 . The
position or displacement 𝒓 = 𝑥𝒆 + 𝑦𝒆 at time 𝑡, where
𝑥 = 𝑣 𝑡 = 𝑣 cos 𝜃 𝑡
𝑦 = 𝑣 𝑡 − 𝑔𝑡 = 𝑣 sin 𝜃 𝑡 − 𝑔𝑡
Eliminate 𝑡 from these two equations gives the trajectory in the form
𝑦 = 𝑥 tan 𝜃 −
sec 𝜃 = sec 𝜃 sin 2𝜃 −
If we set the unit of length of the system as ℎ, then 𝑥 = 𝑥 ℎ, 𝑦 = 𝑦ℎ, and
𝑦 = 𝑥 tan 𝜃 −
sec 𝜃 = sec 𝜃 sin 2𝜃 −
,𝛼 =
This is the same as
𝑥=
sin 2𝜃 ± sin 2𝜃 −
cos 𝜃 = 𝛼 cos 𝜃 sin 𝜃 ± sin 𝜃 −
For a given 𝑣 and 𝑔, we can set ℎ = 𝑣 /𝑔 such that 𝛼 = 1.
We can call this ℎ the characteristic length of the system.
For a parabola, (1) every 𝑥 corresponds to one 𝑦, i.e., a line of constant 𝑥 will meet
the parabola at one and only one point. (2) However, for a 𝑦, it may correspond to
two, one or zero 𝑥, i.e., a line of constant 𝑦 will meet the parabola at two points, one
point or none. The left figure shows the dependence of 𝑦 on 𝜃 for a given 𝑥 in case
(1). The right figure shows the dependence of 𝑥± on 𝜃 for a given 𝑦 in case (2).
𝑥 = 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4,1.6, 1.8
𝑦 = −1.0, −0.8, −0.6, −0.4, −0.2, 0.0, 0.2, 0.4
𝑥 in blue, 𝑥 in orange