Problem P. 5283. (January 2021)
P. 5283. Three friends Sebi, Tóni and Zoli entered for the school's running competition held on Challenge Day. All of them covered the 2.4 km distance at a constant speed. When Tóni just covered 68% of the distance, Sebi had another three minutes to run. Zoli covered 20 cm more in each second than Sebi did, while he covered 10 cm less in each second than Tóni did.
\(\displaystyle a)\) How much time elapsed between the moments when Zoli and Tóni reached the finish line?
\(\displaystyle b)\) How far was Sebi from the finish line when Tóni reached it?
(4 pont)
Deadline expired on February 18, 2021.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. Jelöljük a fiúk sebességét rendre \(\displaystyle V_S\)-sel, \(\displaystyle v_T\)-vel és \(\displaystyle v_Z\)-vel és számoljunk SI-egységekkel (m, m/s, s). A megadott feltételek szerint
| \(\displaystyle (1)\) | \(\displaystyle v_Z=v_S+0{,}2,\) |
| \(\displaystyle (2)\) | \(\displaystyle v_Z=v_T-0{}1.\) |
Tóni \(\displaystyle \frac{2400\cdot 0{,}68}{v_T}\) idő alatt éri el a táv 68%-át, ennél 180 másodperccel hosszabb idő alatt érkezik Sebi a célba. Felírhatjuk tehát, hogy
| \(\displaystyle (3)\) | \(\displaystyle \frac{1632}{v_T}+180=\frac{2400}{v_S}.\) |
Az (1), (2) és (3) egyenletrendszer meghatározza az ismeretlen sebességeket:
\(\displaystyle v_S=4{,}83~\frac{\rm m}{\rm s}, \qquad v_T=5{,}13~\frac{\rm m}{\rm s}, \qquad v_Z=5{,}03~\frac{\rm m}{\rm s}.\)
\(\displaystyle a)\) Zoli és Tóni célba érkezése között
\(\displaystyle \frac{2400}{5{,}03}-\frac{2400}{5{,}13}\approx 9{,}3~\rm s\)
idő telt el.
\(\displaystyle b)\) Tóni 2400/5,13=468 másodperc alatt ért be a célba. Sebi ezalatt 2260 métert tett meg, tehát a céltól még 140 m távol volt.
Statistics:
78 students sent a solution. 4 points: 53 students. 3 points: 10 students. 2 points: 6 students. 1 point: 2 students. 0 point: 1 student. Unfair, not evaluated: 6 solutionss.
Problems in Physics of KöMaL, January 2021