I am working on this problem.
Two men, A and B run in the same direction around a circular track, starting at the same time from the same point. The speed of A is 5 km/hr and that of B is 3 km/hr. If the track has a perimeter of 6 km, after how much time will the two meet?
How to solve this question without using the concept of the relativity?
I believe the answer is 3.
On
To determine when the two men, A and B, will meet, we need to consider their relative speeds and the distance they need to cover to meet.
The relative speed between A and B is the difference in their speeds, which is $5 km/hr - 3 km/hr = 2 km/hr$. To find the time it takes for them to meet, we need to calculate the time it takes to cover the total distance of $6 km$ at a relative speed of $2 km/hr$.
$Time = Distance / Speed$
$Time = 6 km / 2 km/hr$
$Time = 3 hours$
Therefore, the two men will meet after $3$ hours of running.
It is approachable using simple modular arithmetic. We just wish to find the minimal positive solution $t$ of:
$$3t \equiv 5t \pmod 6$$
Which is solved when $3t - 5t = 6z$ for $z \in \mathbb{Z}$. It follows:
$3t - 5t = -2t = 6z \rightarrow t = -3z$ for $z \in \mathbb{Z}$.
We consequently have the first nontrivial positive solution is 3.
Note: if you are unfamiliar with "mod", it is akin to the remainder under integer division. What is occurring mathematically here then is essentially a "reset" when one makes a lap around the track, allowing us to evaluate the problem without relativity.