Chasing Across the Field

Solution & Explanation

Problem Understanding:

In this problem, we have a scenario where a man and his dog are running towards each other on a 100-foot-long football field. The man's speed is $x$ ft/s, while the dog runs at twice the man's speed, i.e., $2x$ ft/s. Each time the dog reaches the man, it turns around, runs back to its starting point, and then returns to the man, repeating this cycle. The task is to calculate the total distance the dog travels by the time the man reaches the opposite end of the field.

Logical Approach:

1. Time Calculation:

   • The man travels the entire length of the football field, which is 100 feet.

   • The time $t$ taken by the man to reach the opposite end is given by:

     $t = \frac{\text{Distance}}{\text{Speed}} = \frac{100}{x} \text{ seconds}$

2. Dog's Movement:

   • The dog runs continuously for the same duration $t$ that the man is running.

   • Since the dog's speed is $2x$, the distance $D_d$ covered by the dog in this time is:

     $D_d = \text{Speed of Dog} \times \text{Time} = 2x \times \frac{100}{x} = 200 \text{ feet}$

Mathematical Justification:

• The dog's speed is twice that of the man's speed. Hence, in the same time period, the dog covers twice the distance as the man.

• The man's total distance covered is 100 feet; therefore, the dog covers:

  $2 \times 100 = 200 \text{ feet}$

Conclusion:

• The dog travels a total distance of 200 feet by the time the man reaches the opposite end of the field.
• The problem can be misleading due to the description of the dog reaching the man and turning around, but the key insight is that the dog runs continuously at twice the man's speed for the entire duration.

Alternative Thought Process:

• The problem can also be framed as a series of harmonic movements where the dog runs back and forth. However, calculating each segment is unnecessary as the continuous running at a constant speed simplifies the calculation.

• The problem demonstrates the importance of understanding relative motion and speeds, rather than being distracted by the dog's back-and-forth movements.

By focusing on the speeds and total time, we can derive the total distance covered efficiently.