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The problem you're asking about involves a runner and a bird. Here's a step-by-step breakdown to solve the first part of the question:

### Given:
- The runner's speed: \( v_r = 2.4 \, \text{km/h} \)
- The bird's speed: \( v_b = 9.6 \, \text{km/h} \) (4 times the runner's speed)
- The distance from the finish line: \( L = 7 \, \text{km} \)

### Problem:
Find the cumulative distance traveled by the bird before the runner reaches the finish line.

### Steps:

1. **Time for the runner to reach the finish line:**
   The time it takes for the runner to cover the 7 km at a speed of 2.4 km/h is calculated as:
   \[
   t_{\text{runner}} = \frac{L}{v_r} = \frac{7}{2.4} \approx 2.92 \, \text{hours}
   \]

2. **Bird's total flight time:**
   Since the bird starts flying at the same time as the runner, the bird will be in motion for the same duration (until the runner reaches the finish line). Therefore, the bird's flight time is also \( t_{\text{bird}} = t_{\text{runner}} = 2.92 \, \text{hours} \).

3. **Cumulative distance traveled by the bird:**
   The bird's speed is \( v_b = 9.6 \, \text{km/h} \). The total distance the bird travels in the 2.92 hours is:
   \[
   d_{\text{bird}} = v_b \times t_{\text{bird}} = 9.6 \times 2.92 \approx 28.03 \, \text{km}
   \]

### Answer:
The cumulative distance traveled by the bird is approximately **28.03 km**.

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Would you like further details on the next part of the problem? Here are five questions to help expand your understanding:

1. How does the relative speed between the bird and the runner affect the bird's total distance?
2. What would happen to the total distance if the runner's speed increased?
3. How can we verify that the bird's turning time is negligible in this problem?
4. What if the bird had a different constant speed — how would that affect the total distance calculation?
5. Can we model this problem using integrals for more complex bird paths?

### Tip:
Always check units consistency when solving motion problems to avoid errors.

A runner is jogging in a straight line at a steady vr = 2.4 km/h. When the runner is L = 7 km from the finish line, a bird begins flying straight from the runner to the finish line at vb = 9.6 km/h (4 times as fast as the runner). What cumulative distance does the bird travel? Assume the bird occupies one point in space (zero length), travels in a straight line, and can turn without loss of speed. Answer in units of km.

Solve a classic kinematics problem involving a runner and a bird traveling at different speeds. Learn how to calculate the cumulative distance traveled by the bird as it flies back and forth while the runner moves towards the finish line. This problem applies concepts of relative speed and uniform motion.

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