| d2x | = 0, | d2y | = −g. |
| dt2 | dt2 |
(3)
The solution is
x = At + B, y = −1⁄2gt2 + Ct + D.
(4)
If the initial values of x, y, ẋ, ẏ are given, we have four conditions to determine the four arbitrary constants A, B, C, D. Thus if the particle start at time t = 0 from the origin, with the component velocities u0, v0, we have
x = u0t, y = v0t − 1⁄2gt2.
(5)
Eliminating t we have the equation of the path, viz.
| y = | v0 | x − | gx2 | . |
| u0 | 2u2 |