Two runners start moving from the same point A on a circular track but in oppostite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at point A again, how many times do they meet, excluding the start and finish?
Student
To solve this problem, we need to first find the time it takes for the runners to meet at point A again. Then, we can determine how many times they meet during that time period.
Let's say the circumference of the circular track is C feet. Since the runners are moving in opposite directions, their combined speed is 5 + 9 = 14 ft. per second.
The time it takes for the runners to meet again can be calculated using the formula:
time = distance / speed
The distance each runner covers before they meet is equal to the circumference of the track. So, the time it takes for them to meet again is:
time = C / 14
Now, to find the number of times they meet, we need to consider the relative speed between the two runners. The relative speed is the difference between their speeds, which is 9 - 5 = 4 ft. per second.
During the time it takes for the runners to meet again, the relative speed will determine how many times they meet. Each time they meet, one runner will have completed one lap more than the other.
Since one lap is equal to the circumference of the track, the number of times they meet is:
number of meetings = relative speed / circumference
Substituting the relative speed and the formula for the time it takes for them to meet again:
number of meetings = (9 - 5) / C / 14
= 4 / C / 14
= 4 * 14 / C
= 56 / C
Therefore, the number of times they meet, excluding the start and finish, is 56 divided by the circumference of the circular track.