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 determine the time difference between the inside and outside runners, we can use the formula for the circumference of a circle:

C = 2πr

The inside lane has a radius of 50 m, so the distance the inside runner has to cover is:

C_inside = 2π(50) = 100π m

The outside lane has a radius of 51 m, so the distance the outside runner has to cover is:

C_outside = 2π(51) = 102π m

Now, we can calculate the time it takes for each runner to complete their respective distances.

Time taken by the inside runner:

Time_inside = Distance_inside / Speed = (100π m) / (7 m/s)

Time taken by the outside runner:

Time_outside = Distance_outside / Speed = (102π m) / (7 m/s)

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

Time_difference = Time_inside - Time_outside = (100π m) / (7 m/s) - (102π m) / (7 m/s)

Simplifying the expression further:

Time_difference = (100π - 102π) / (7 m/s) = -2π / (7 m/s)

Therefore, the inside runner will finish the race approximately -2π / (7 m/s) seconds ahead of the outside runner. Note that the negative sign indicates that the outside runner finishes before the inside runner, given the unequal lane lengths on the curve.

To find out by how many seconds the inside runner will beat the outside runner, we need to determine the distance traveled by each runner.

To calculate the distance, we can use the formula:

distance = circumference of the circle

The circumference of a circle is given by the formula:

circumference = 2 * π * radius

Given that the radius of the inside lane is 50 m, we can calculate its circumference as:

circumference of inside lane = 2 * π * 50

Next, we can calculate the time taken for the inside runner to complete one lap by dividing the distance by the speed:

time taken for inside runner = distance / speed

Substituting the values, we get:

time taken for inside runner = (2 * π * 50) / 7

Similarly, we can calculate the time taken for the outside runner by considering the circumference of the outside lane, which is 51 m:

time taken for outside runner = (2 * π * 51) / 7

To find the time difference, we can subtract the time taken by the outside runner from the time taken by the inside runner:

time difference = time taken for inside runner - time taken for outside runner

Substituting the values and simplifying, we get:

time difference = [(2 * π * 50) / 7] - [(2 * π * 51) / 7]

Now we can evaluate this expression to find the time difference in seconds.