Jon begins jogging at a steady 3m/sec down the middle of lane one of a public track. Laura starts even with him in the center of lane two but moves 4m/sec. At the instant they begin, Ellie is located 100 meters down he track in lane four, and is heading towards them in his lane at 6m/sec. After how many seconds will the runners lie in a straight line?

8 and 1/3 sec

Well, if they're all running in straight lines, they're not really lying, are they? But I get what you mean!

Let's think this through. Jon is running at 3m/sec, Laura is running at 4m/sec, and Ellie is running at 6m/sec.

To find the time it takes for them to be in a straight line, we need to calculate the distance traveled by each person.

Let's assume that the time it takes for them to be in a straight line is 't' seconds.

Jon will have traveled a distance of 3t meters, Laura will have traveled a distance of 4t meters, and Ellie will have traveled a distance of 6t meters.

To be in a straight line, the total distance traveled by all three of them should be equal.

So, we have the equation: 3t + 4t + 100 (the distance Ellie is starting from) = 6t

Simplifying this equation, we get: 7t + 100 = 6t

So, t = 100 seconds.

Therefore, after 100 seconds, Jon, Laura, and Ellie will be in a straight line.

To find out when the runners will lie in a straight line, we need to determine the time it takes for each runner to travel the distance between their starting point and the point where they are in a straight line.

Let's assume the time it takes for all the runners to be in a straight line is "t" seconds.

The distance Jon travels in "t" seconds will be: Distance = Speed x Time = 3t meters.

The distance Laura travels in "t" seconds will be: Distance = Speed x Time = 4t meters.

The distance Ellie travels in "t" seconds will be: Distance = Speed x Time = 6t meters.

At the point when they are in a straight line, the sum of the distances traveled by the runners from their starting points will be equal to the distance between Jon's starting point and Ellie's starting point. In this case, it is 100 meters.

So, we can set up the following equation:
3t + 4t + 6t = 100

Combining like terms:
13t = 100

Dividing both sides by 13:
t = 100/13

So, the runners will lie in a straight line after approximately 7.69 seconds.

To find out how many seconds it will take for the runners to lie in a straight line, we need to consider their relative motion and when they will align.

Let's first determine Laura's position relative to Jon's position. Since Laura is moving faster than Jon, she will catch up to him. The rate at which Laura is catching up to Jon is the difference in their speeds, which is 4m/sec - 3m/sec = 1m/sec.

To calculate the time it takes for Laura to catch up to Jon, we can use the equation:

Distance = Speed × Time

Since Jon starts at the beginning of the track and Laura starts with a 100-meter head start, the distance between them when Laura catches up to Jon is 100 meters. Plugging in the values, we get:

100m = 1m/sec × Time

Simplifying the equation:

Time = 100m / 1m/sec
Time = 100 seconds

So, it will take 100 seconds for Laura to catch up to Jon.

Now, let's consider Ellie's position relative to Jon's position. Since Ellie is moving towards Jon, the relative speed between Ellie and Jon is the sum of their speeds, which is 3m/sec + 6m/sec = 9m/sec.

To calculate the time it takes for Ellie to reach Jon, we can again use the equation:

Distance = Speed × Time

Since Ellie is 100 meters away from Jon, plugging in the values, we get:

100m = 9m/sec × Time

Simplifying the equation:

Time = 100m / 9m/sec
Time ≈ 11.11 seconds

So, it will take approximately 11.11 seconds for Ellie to reach Jon.

Therefore, the total time it will take for the runners to lie in a straight line is the time it takes for Laura to catch up to Jon, which is 100 seconds, plus the time it takes for Ellie to reach Jon, which is approximately 11.11 seconds.

Total Time = 100 seconds + 11.11 seconds
Total Time ≈ 111.11 seconds

Thus, it will take approximately 111.11 seconds for the runners to lie in a straight line.