Two friends are at the local high school track, a circle measuring 440 yards for one complete lap. Abe can jog at 8.2 miles per hour while Bob’s jogging speed is 4.6 miles per hour. If they both start at the same point and jog in the same direction (say clockwise), how many times would they have crossed each other after a half hour? If they started off in opposite directions, what would your answer be?

s = 440 yards = 402.3 m

v1 = 8.2 mph = 3.7 m/s,
v2 = 4.6 mph = 2.1 m/s.

Imagine that B is at rest, then A is jogging at the relative velocity
v = v1-v2 =3.7 – 2.1 = 1.6 m/s.
The time of 1 lap motion is
T = s/v=402.3/1.6 =251 s.
The number if their meetings is
N = t/T =0.5•3600/251 =7.17 => 7.
When they are moving un opposite directions the relative velocity is
v = v1+v2 = 3.7 +2.1 = 5.8 m/s.
T = s/v =402.3/5.8 =69.3 s.
N =t/T =0.5•3600/69.3 = 25.9 => 25

Thanks a lot Elena

To find out how many times Abe and Bob cross each other, we need to determine the number of laps they complete in a given time.

First, let's convert their jogging speeds to yards per minute, which will help us calculate the rate at which they cover the track.

Abe's speed: 8.2 miles/hour = (8.2 * 1760 yards) / 60 minutes = 299.2 yards/minute

Bob's speed: 4.6 miles/hour = (4.6 * 1760 yards) / 60 minutes = 167.2 yards/minute

Now, let's consider the scenario where they start at the same point and jog in the same direction (clockwise):

Since they both jog in the same direction, their relative speed will be the difference between their individual speeds. In this case, the relative speed is: 299.2 - 167.2 = 132 yards/minute.

In a half hour (30 minutes), Abe would've covered a distance of 299.2 * 30 = 8976 yards.

Similarly, Bob would've covered a distance of 167.2 * 30 = 5016 yards.

Now, we can divide their respective distances by the length of the track (440 yards) to find the number of laps completed:

Number of laps Abe completes: 8976 / 440 ≈ 20.4 laps

Number of laps Bob completes: 5016 / 440 ≈ 11.4 laps

Since they both start at the same point, they will cross each other each time Abe completes one lap more than Bob.

Therefore, they would have crossed each other 20 times (since Bob completes 11 complete laps) after a half hour if they started off in the same direction.

If they started off in opposite directions (counter-clockwise):

Their relative speed would be the sum of their individual speeds: 299.2 + 167.2 = 466.4 yards/minute.

In this case, since their relative speed is greater than the track length, they will never cross each other.

Hence, if they started off in opposite directions, they would never cross each other in any given time interval.