If you ever watch track and field at the Olympics, you'll notice that the starting points of the runners are not even if the race is run on a curve and the runners have to run in lanes. This is to offset the fact that the distances along each lane are unequal on a curve. To see this effect, consider two runners, each of whom runs at a speed of 7 m/s. The runners run on a circular track. The radius of the inside lane is 50 m, and the radius of the outside lane is 51 m. By how many seconds will the inside runner beat the outside runner if they each run once around the track?

To solve this problem, we can use the formula for calculating the circumference of a circle: C = 2πr, where C is the circumference and r is the radius.

1. Calculate the circumference of the inside and outside lanes:
- Circumference of inside lane = 2π(50) = 100π m
- Circumference of outside lane = 2π(51) = 102π m

2. Calculate the time it takes for each runner to complete one lap by dividing the distance (circumference) by their speed:
- Time for inside runner = (100π m) / (7 m/s) = (100/7)π s
- Time for outside runner = (102π m) / (7 m/s) = (102/7)π s

3. Find the difference in time between the two runners by subtracting the time for the inside runner from the time for the outside runner:
- Difference in time = (102/7)π s - (100/7)π s = (2/7)π s

Therefore, the inside runner will beat the outside runner by (2/7)π seconds, or approximately 0.902 seconds.

To find the time difference between the inside runner and the outside runner, we need to calculate the distance each runner runs.

The distance the inside runner runs is the circumference of the circle with radius 50m, which can be found using the formula C = 2πr. Therefore, the distance is:

C_inside = 2π * 50 = 100π meters.

Similarly, the distance the outside runner runs is the circumference of the circle with radius 51m:

C_outside = 2π * 51 = 102π meters.

Now we can calculate the time it takes for each runner to complete their respective distances. The time can be found using the formula: time = distance / speed.

For the inside runner:
time_inside = C_inside / 7

For the outside runner:
time_outside = C_outside / 7

Substituting the values, we get:

time_inside = (100π) / 7 ≈ 14.29π seconds
time_outside = (102π) / 7 ≈ 14.57π seconds

To find the time difference between the two runners, we subtract the time taken by the outside runner from the time taken by the inside runner:

time_difference = time_inside - time_outside
= 14.29π - 14.57π
≈ -0.28π seconds.

Therefore, the inside runner will beat the outside runner by approximately 0.28π seconds.