To solve this problem, we'll use the concept of relative motion and uniform distribution of trains.
Let's denote:
vv as the speed of the trains (same in both directions).
TbTb as the time interval between trains going back to the city (from dacha to city).
D=v×TbD=v×Tb as the distance between two consecutive trains moving towards the city.
n=1D=1v×Tbn=D1=v×Tb1 as the number of trains per unit length heading towards the city.
When you are traveling towards the dacha (from city to dacha) at speed vv, you meet oncoming trains every 5 minutes. The relative speed between your train and the oncoming trains is v+v=2vv+v=2v.
The rate at which you meet the oncoming trains is:
ChatGPT:
To solve this problem, we'll use the concept of relative motion and uniform distribution of trains.
Let's denote:
vv as the speed of the trains (same in both directions).
TbTb as the time interval between trains going back to the city (from dacha to city).
D=v×TbD=v×Tb as the distance between two consecutive trains moving towards the city.
n=1D=1v×Tbn=D1=v×Tb1 as the number of trains per unit length heading towards the city.
When you are traveling towards the dacha (from city to dacha) at speed vv, you meet oncoming trains every 5 minutes. The relative speed between your train and the oncoming trains is v+v=2vv+v=2v.
The rate at which you meet the oncoming trains is:
Meeting rate=Relative speed×Train density=(2v)×1vTb=2TbMeeting rate=Relative speed×Train density=(2v)×vTb1=Tb2
Given that trains are met every 5 minutes, the meeting rate is:
Meeting rate=15 minutesMeeting rate=5 minutes1
Setting the two expressions for the meeting rate equal to each other:
2Tb=15Tb2=51
Tb=10 minutesTb=10 minutes
This means trains heading back to the city depart every 10 minutes. Therefore, the number of trains arriving in the city in one hour is:
Number of trains per hour=60 minutesTb=6010=6Number of trains per hour=Tb60 minutes=1060=6
Answer: 6