To solve this problem, we can use the concept of relative speed. Relative speed is the speed at which one object is moving with respect to another object.
Let's denote:
- \(d\) as the distance traveled by both men before the second man overtakes the first,
- \(t\) as the time taken for the second man to overtake the first.
For the first man:
- Speed = 5 km/h
- Time = \(t + 6\) hours (since he has already traveled for 6 hours when the second man starts)
For the second man:
- Speed = 8 km/h
- Time = \(t\) hours
Since distance = speed * time for both men, we have:
For the first man: \(d = 5(t + 6)\)
For the second man: \(d = 8t\)
Since both men cover the same distance when the second man overtakes the first, we can set these two equations equal to each other:
\[5(t + 6) = 8t\]
Now, let's solve for \(t\):
\[5t + 30 = 8t\]
\[30 = 8t - 5t\]
\[30 = 3t\]
\[t = \frac{30}{3} = 10\]
So, it will take the second man 10 hours to overtake the first man.
To find out when this happens, we can substitute \(t = 10\) into either of the equations:
For the first man: \(d = 5(t + 6) = 5(10 + 6) = 5(16) = 80\) km
For the second man: \(d = 8t = 8(10) = 80\) km
So, both men will be at the same point after 10 hours, at a distance of 80 km from their starting point.
