Problem P. 5262. (November 2020)

P. 5262. Formula One car drivers are participating in a race in which reaching the greatest speed is not the best tactic to win. A designated distance of \(\displaystyle d = 1250\) m is to be covered at a constant speed, then each car has to stop at a deceleration of \(\displaystyle a = 2~\mathrm{m/s}^2\). The winner is the driver who can stop in the least time, measured from the start of the car.

\(\displaystyle a)\) What should the speed of the winning car be at the constant speed stage of the motion, if the driver wants to stop in the least time?

\(\displaystyle b)\) How much distance does the winning car cover in this case from the start to the stop?

(4 pont)

Deadline expired on December 15, 2020.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. \(\displaystyle a)\) Állandó \(\displaystyle v\) sebességgel \(\displaystyle d\) hosszúságú utat

\(\displaystyle t_1=\frac{d}{v}\)

idő alatt tesz meg egy autó. \(\displaystyle v\) sebességről \(\displaystyle a\) lassulással

\(\displaystyle t_2=\frac{a}{v}\)

idő alatt tud megállni. A teljes időtartam (a számtani és a mértani középre vonatkozó egyenlőtlenség szerint)

\(\displaystyle T=t_1+t_2=\frac{d}{v}+\frac{v}{a}\ge 2\sqrt{\frac{d}{v}\cdot\frac{v}{a}}=2\sqrt{\frac{d}{a}}=50~\rm s.\)

Optimális esetben \(\displaystyle t_1=t_2=25~\rm s\), vagyis

\(\displaystyle \frac{d}{v}=\frac{v}{a}, \qquad \text{azaz}\qquad v=\sqrt{ad}=50~\frac{\rm m}{\rm s}=180~\frac{\rm km}{\rm h}.\)

\(\displaystyle b)\) A teljes út hossza a legjobb sebességválasztás esetén

\(\displaystyle s=vt_1+\frac{a}{2}t_2^2=1250~{\rm m}+625~{\rm m}=1875~{\rm m}.\)

Statistics:

102 students sent a solution.
4 points:	69 students.
3 points:	13 students.
2 points:	4 students.
1 point:	6 students.
Unfair, not evaluated:	10 solutionss.

Problems in Physics of KöMaL, November 2020