Determining the Time for a Faster Car to Double the Distance Ahead of a Slower Car

In a scenario where two cars are traveling in the same direction, determining the time it will take for one car to double the distance ahead of another car can be a valuable exercise in understanding relative speed and time calculations. This article delves into the specifics of this scenario, offering a step-by-step solution to the problem. For SEO optimization, key terms such as 'distance ahead,' 'relative speed,' and 'time calculation' have been chosen to enhance relevance and visibility on search engines.

Scenario

Consider two cars, Car A and Car B, traveling in the same direction. Initially, Car A is 18 miles ahead of Car B. Car A travels at a speed of 55 mph, while Car B travels at a speed of 45 mph.

Steps to Solve the Problem

1. **Initial Distance:**
- Car A: 18 miles ahead of Car B. 2. **Speed of Each Car:**
- Speed of Car A: 55 mph - Speed of Car B: 45 mph 3. **Relative Speed:**
- Relative speed at which Car A is pulling away from Car B is given by:
Relative speed Speed of Car A - Speed of Car B 55 mph - 45 mph 10 mph 4. **Target Distance:**
- To double the distance, the target distance is 36 miles (2 times 18 miles). 5. **Additional Distance Needed:**
- Additional distance needed for Car A to pull away from Car B is 36 miles - 18 miles 18 miles.

Time Calculation

The time it takes for Car A to increase the distance by 18 miles can be calculated using the relative speed. The formula used is: Time Distance / Relative Speed By substituting the values, we get: Time 18 miles / 10 mph 1.8 hours

Thus, it will take approximately 1.8 hours for Car A to double the distance between itself and Car B.

Alternative Mathematical Approach

Using an algebraic approach, let t be the time before Car A is 36 miles ahead of Car B. The equation derived is:

Solving for t using algebraic method:
   45t   36 - 18  55t
   45t   18  55t
   18  10t
   t  18 / 10
   t  1.8 hours

This confirms that it will take 1.8 hours before Car A is 36 miles ahead of Car B.

Conclusion

Understanding and applying the concept of relative speed and time calculation is crucial in scenarios involving distances and speeds. This problem is a classic example of how to determine the time it takes for one moving object to increase its distance ahead of another. The step-by-step solution demonstrates the practical application of mathematical principles in real-world scenarios.

Key Terms

Distance ahead: The initial lead one car has over another.

Relative speed: The speed at which one object is moving away from or towards another object.

Time calculation: Using given speeds and distances to determine the time taken for an event to occur.