005 (part 1 of 2) 10.0 points A runner is jogging in a straight line at a steady vr= 6.2 km/hr. When the runner is L= 5.7 km from the finish line, a bird begins flying straight from the runner to the finish line at vb= 24.8 km/hr (4 times as fast as the runner). When the bird reaches the finish line, it turns around and flies directly back to the runner. L vb vr finish line What cumulative distance does the bird travel? Even though the bird is a dodo, assume that it occupies only one point in space (a “zero” length bird), travels in a straight line, and that it can turn without loss of speed. Answer in units of km

To solve this problem, we need to determine the total distance the bird travels while the runner is jogging towards the finish line.

### Step-by-Step Solution

1. **Calculate the time taken by the runner to reach the finish line:**

   The runner is jogging at a speed of \( v_r = 6.2 \) km/hr, and is \( L = 5.7 \) km away from the finish line.

   The time \( t \) taken by the runner to reach the finish line can be calculated by dividing the distance \( L \) by the speed \( v_r \):

   \[
   t = \frac{L}{v_r} = \frac{5.7 \, \text{km}}{6.2 \, \text{km/hr}}
   \]

   Let's calculate this time:

   \[
   t = \frac{5.7}{6.2} \approx 0.9194 \, \text{hr}
   \]

2. **Calculate the distance the bird travels:**

   The bird starts flying towards the finish line at a speed of \( v_b = 24.8 \) km/hr. During the entire time that the runner is jogging to the finish line (which is \( t = 0.9194 \) hours), the bird keeps flying back and forth between the runner and the finish line.

   Since the bird flies for the entire duration \( t \) at a constant speed \( v_b \), the cumulative distance \( D_b \) the bird travels is simply its speed multiplied by the time:

   \[
   D_b = v_b \times t
   \]

   Plugging in the values:

   \[
   D_b = 24.8 \, \text{km/hr} \times 0.9194 \, \text{hr}
   \]

   Let's calculate this distance:

   \[
   D_b = 24.8 \times 0.9194 \approx 22.80 \, \text{km}
   \]

### Final Answer

The cumulative distance that the bird travels is **22.80 km**.

Would you like more details on any step or have any other questions?

Here are 5 related questions for further exploration:

1. What would the distance traveled by the bird be if its speed was 30 km/hr instead of 24.8 km/hr?
2. How would the distance traveled by the bird change if the runner’s speed increased to 8 km/hr?
3. If the distance from the runner to the finish line was doubled, how would that affect the bird’s total distance traveled?
4. How would the problem change if the bird had a constant acceleration instead of a constant speed?
5. Can you calculate the average speed of the bird during the journey?

**Tip:** In problems involving relative motion, the key is often to find the time each object spends in motion and multiply it by their speed to find the distance traveled.

Explore how to calculate the cumulative distance traveled by a bird flying back and forth between a runner and the finish line. This problem involves relative motion and requires speed and time calculations. Discover the solution step-by-step with detailed calculations and insights.

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