A runner is jogging at a steady 8.4 km/hr.

When the runner is 2.6 km from the finish
line, a bird begins flying from the runner to
the finish line at 42 km/hr (5 times as fast
as the runner). When the bird reaches the
finish line, it turns around and flies back to
the runner.
L
vb
vr
finish
line
How far does the bird travel? Even though
the bird is a dodo, assume that it occupies
only one point in space (a “zero” length bird)
and that it can turn without loss of speed.
Answer in units of km

This is my take on it :Flight time of bird to finish line FL = 2.6/42 =(approx) .06hr
Distance runner travels during time period = .06(8.4) = .52km
Distance of runner to FL = 2.6-.52 = 2.08km
Net closing speed between runner and bird = 8.4km +42 km = 50.4km
Time for runner and bird to meet 2.6/50.4 = (approx) .05hr
Distance covered by runner (.05)8.4 = (approx) .43 km
Distance covered by bird to runner (.05)(42) = (approx) 2.1km

2.1 + 2.6 = 4.7km (its not correct) :( What am I doing wrong?

d = Vt = 2.6 km.

42t = 2.6,
t = 2.6/42 = 0.062 h. = Time for bird to reach finish line.

d = Vt=8.4 * 0.062 = 0.521 km=Distance
covered by runner as bird traveled to finish line.

Dr = 2.6 - 0.52 = 2.08 km = Runner's distance when bird reached finish line.

dr + db = 2.08 km,
8.4t + 42t = 2.08,
50.4t = 2.08,
t = 2.08/50.4 = 0.04127 h.

db = Vt = 42 * 0.04127 = 1.733 km. =
Dist. traveled by bird while meeting runner.

Db = 2.6 + 1.733 = 4.33 km.=Tot. dist.
traveled by bird.

Even tho your approach was different from mine, I believe it was correct.
But you made one error: 2.6/50.4 shouled be 2.08/50.4.

Well, I hate to break it to you, but it seems like you've been running in circles with your calculations. Let me help you untangle this mess and find the correct answer.

We know that the bird is 2.6 km away from the finish line when it starts flying towards it. The bird is faster than the runner, so it will eventually reach the finish line and then turn around.

To find the distance traveled by the bird, we need to calculate how long it takes for the bird to reach the finish line. We know that the bird's speed is 42 km/hr, so the time it takes to reach the finish line is:

Time = Distance / Speed = 2.6 km / 42 km/hr = 0.062 hr

Now, during this time, the runner keeps running at a speed of 8.4 km/hr. So, the distance covered by the runner while the bird is flying is:

Distance runner covers = Speed × Time = 8.4 km/hr × 0.062 hr = 0.52 km

Now, the bird has reached the finish line and starts flying back towards the runner. The total distance covered by the bird is twice the distance from the finish line to the runner:

Distance covered by bird = 2 × (2.6 km - 0.52 km) = 2 × 2.08 km = 4.16 km

So, the bird travels a total distance of 4.16 km. And remember, despite being a dodo, it still managed to fly to and fro without any loss of speed! Quite impressive for a bird with zero length, don't you think?

Your calculations for the flight time of the bird to the finish line and the distance traveled by the runner during that time are correct.

However, the distance of the runner to the finish line after the bird reaches it should be 2.6 km - 0.52 km = 2.08 km instead of 2.6 km - 0.43 km = 2.17 km.

The net closing speed between the runner and the bird is indeed 8.4 km/h + 42 km/h = 50.4 km/h.

But the time for the runner and the bird to meet is 2.08 km / 50.4 km/h ≈ 0.0413 hours (instead of 2.6 km / 50.4 km/h = 0.0516 hours).

The distance covered by the runner during this time is 0.0413 hours * 8.4 km/h ≈ 0.345 km (instead of 0.05 * 8.4 km = 0.43 km).

And the distance covered by the bird to reach the runner is 0.0413 hours * 42 km/h ≈ 1.746 km (instead of 0.05 * 42 km = 2.1 km).

Adding up the distance traveled by the bird to reach the finish line (2.6 km) and the distance traveled by the bird to reach the runner (1.746 km), the total distance traveled by the bird is approximately 4.346 km.

To correctly calculate the distance the bird travels, we need to consider the motion of both the runner and the bird. Let's break down the problem step by step:

1. First, let's find the time it takes for the bird to reach the finish line from the initial position of the runner. The distance between the runner and the finish line is 2.6 km, and the bird's speed is 42 km/hr. Therefore, the time it takes for the bird to reach the finish line is 2.6 km / 42 km/hr ≈ 0.062 hr.

2. During this time, the runner continues to jog. The distance the runner travels during this time can be found by multiplying the runner's speed (8.4 km/hr) by the time (0.062 hr). Thus, the distance covered by the runner is 8.4 km/hr × 0.062 hr ≈ 0.52 km.

3. After the bird reaches the finish line, it turns around and flies back towards the runner. At this point, the runner is 0.52 km away from the finish line.

4. Now let's consider the time it takes for the runner and the bird to meet. The combined speed of the runner and the bird is the sum of their individual speeds, which is 8.4 km/hr + 42 km/hr = 50.4 km/hr. The distance between the runner and the bird is 0.52 km.

5. By dividing the distance (0.52 km) by the combined speed (50.4 km/hr), we can determine the time it takes for the runner and the bird to meet: 0.52 km / 50.4 km/hr ≈ 0.0103 hr.

6. During this time, the bird will travel its own speed (42 km/hr) multiplied by the time (0.0103 hr). Therefore, the distance covered by the bird to meet the runner is 42 km/hr × 0.0103 hr ≈ 0.4336 km.

7. Finally, to find the total distance the bird travels, we add the distance from the initial position to the finish line (2.6 km), plus the distance from the finish line back to the runner (0.52 km), plus the distance covered by the bird to meet the runner (0.4336 km). This gives us a total distance of 2.6 km + 0.52 km + 0.4336 km ≈ 3.5536 km.

So, the bird travels approximately 3.5536 km.