From point $A$, we launch a heavy object vertically upwards with an initial velocity of $v_{0}$. At the same time, from point $B$, which is $a$ meters above $A$, we drop another heavy object with no initial velocity. After how many seconds will the two objects meet? How much distance will each have traveled? Determine the conditions under which the two objects meet:

1.) Between points $A$ and $B$,

2.) At point $A$,

3.) Below point $A$. For example, $v_{0}=30 \mathrm{~m} / \mathrm{sec}, \quad \mathrm{a}=420 \mathrm{~m}, \mathrm{~g}=9.8 \mathrm{~m} \mathrm{sec}^{-2}$.
The two bodies meet after $t$ seconds; during this time, the launched body travels $v_{0} t-\frac{1}{2} g t^{2}$, and the dropped body travels $\frac{1}{2} g t^{2}$, the sum of which is $a$, that is,

$$
v_{0} t-\frac{1}{2} g t^{2}+\frac{1}{2} g t^{2}=a
$$

from which

$$
t=\frac{a}{v_{0}}
$$

During this time, $A$ travels $s_{1}=a\left(1-\frac{a g}{2 v_{0}^{2}}\right)$ and $B$ travels $s_{2}=\frac{1}{2} g \frac{a^{2}}{v_{0}^{2}}$. $t$ and $s_{2}$ are always positive, but $s_{1} \lesseqgtr 0$.

1.) If $a<\frac{2 v_{0}^{2}}{g}$, then $s_{1}>0$; the two bodies meet between $A$ and $B$.

Since $A$ rises for $\frac{v_{0}}{g}$ seconds, if $t<\frac{v_{0}}{g}$, the meeting occurs while $A$ is ascending. In the first case, $a<\frac{v_{0}^{2}}{g}$. If $t>\frac{v_{0}}{g}$, the meeting occurs when $A$ is descending. In the first case, $a>\frac{v_{0}^{2}}{g}$. If $t=\frac{v_{0}}{g}$, or $a=\frac{v_{0}^{2}}{g}$, the two bodies meet at the midpoint of the path; in this case, $s_{1}=\frac{v_{0}^{2}}{2 g}, s_{2}=\frac{v_{0}^{2}}{2 g}$.

2.) If $a=\frac{2 v_{0}^{2}}{g}$, then $s_{1}=0$; the two bodies meet at $A$.

3.) If $a>\frac{2 v_{0}^{2}}{g}$, then $s_{1}<0$; the two bodies meet below $A$.

Given the provided numerical values, we find that $t=14 \mathrm{~s} . s_{2}=960.4 \mathrm{~m}$; the two bodies meet $540.4 \mathrm{~m}$ below $A$.

The problem was solved by: Bálint Béla, Friedmann Bernát, Goldziher Károly, Grünhut Béla, Hofbauer Ervin, Kántor Nándor, Riesz Frigyes, Schólcz Károly, Schvarcz Endre (Székesfehérvár), Szabó Károly, Thiringer Aurél.