From point $A$, we launch a heavy body at an angle $\alpha$ with an initial velocity $c$. Determine the equation of the trajectory with respect to the axes of the direction of the launch and the direction of the gravitational force. What is the geometric significance of the constant appearing in the equation?

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From point $A$, we launch a heavy body at an angle $\alpha$ with an initial velocity $c$. Determine the equation of the trajectory with respect to the axes of the direction of the launch and the direction of the gravitational force. What is the geometric significance of the constant appearing in the equation?
If a body were to rise under the influence of the velocity $c$, the distance from point $A$ would be $A B=c t$.

If the body were under the influence of gravity alone, it would reach a point $C$ below in the same time, such that $A C=\frac{g t^{2}}{2}$.

If, however, it proceeds under the influence of both, after time $t$ it will be at a point $D$, whose coordinates in the adopted system are

$$
\begin{gathered}
x=c t \\
y=\frac{g t^{2}}{2}
\end{gathered}
$$

thus the equation of the trajectory curve is:

$$
x^{2}=\frac{2 c^{2}}{g} y
$$

Since it is known that the equation of a parabola, referred to a tangent and the diameter drawn from the point of tangency as coordinate axes, is

$$
x^{2}=2 p y
$$

it follows that

$$
\begin{aligned}
\frac{2 c^{2}}{g} & =2 p \\
\frac{c^{2}}{g} & =p
\end{aligned}
$$

that is, the constant in question is the parameter of the parabola.

(Aladár Visnya, Vol. VIII, n.d., Pécs.)