Ben Starts Walking Along A Path At $4 \, \text{mi/h}$. One And A Half Hours After Ben Leaves, His Sister Amanda Begins Jogging Along The Same Path At $6 \, \text{mi/h}$. How Long Will It Be Before Amanda Catches Up To Ben?
Introduction
In this article, we will delve into a classic problem of relative motion, where Ben and Amanda engage in a thrilling chase along a path. Ben starts walking at a leisurely pace of 4 miles per hour, while Amanda begins jogging at a faster speed of 6 miles per hour. The question on everyone's mind is: how long will it take for Amanda to catch up to Ben? In this discussion, we will employ mathematical concepts to unravel the mystery behind their speedy chase.
The Problem
Ben starts walking along a path at a speed of 4 miles per hour. One and a half hours after Ben leaves, his sister Amanda begins jogging along the same path at a speed of 6 miles per hour. We need to determine the time it will take for Amanda to catch up to Ben.
Mathematical Formulation
To solve this problem, we will use the concept of relative motion. We will first calculate the distance Ben travels in 1.5 hours, and then determine the time it takes for Amanda to cover the same distance. Let's denote the time it takes for Amanda to catch up to Ben as t hours.
Distance Traveled by Ben
Ben travels at a speed of 4 miles per hour for 1.5 hours. To calculate the distance he travels, we multiply his speed by the time:
distance_ben = speed_ben * time_ben
distance_ben = 4 * 1.5
distance_ben = 6 miles
Relative Speed
When Amanda starts jogging, she is 1.5 hours behind Ben. To catch up to him, she needs to cover the distance he has traveled in 1.5 hours, plus the distance she travels in time t. The relative speed between Amanda and Ben is the difference between their speeds:
relative_speed = speed_amanda - speed_ben
relative_speed = 6 - 4
relative_speed = 2 miles per hour
Time to Catch Up
Now, we can use the relative speed to determine the time it takes for Amanda to catch up to Ben. We know that Amanda needs to cover the distance Ben has traveled in 1.5 hours, plus the distance she travels in time t. We can set up an equation based on this:
distance_ben + relative_speed * t = distance_ben + speed_amanda * t
Simplifying the equation, we get:
relative_speed * t = speed_amanda * t - distance_ben
Now, we can solve for t:
t = distance_ben / (speed_amanda - relative_speed)
t = 6 / (6 - 2)
t = 6 / 4
t = 1.5 hours
Conclusion
Introduction
In our previous article, we explored the problem of Ben and Amanda's speedy chase along a path. Ben starts walking at a leisurely pace of 4 miles per hour, while Amanda begins jogging at a faster speed of 6 miles per hour. We determined that it would take Amanda 1.5 hours to catch up to Ben. In this article, we will answer some frequently asked questions related to this problem.
Q&A
Q: What if Ben and Amanda start at the same point?
A: If Ben and Amanda start at the same point, the problem becomes much simpler. In this case, Amanda would catch up to Ben in the time it takes her to cover the distance Ben travels in 1.5 hours. Since Ben travels at 4 miles per hour, he would cover a distance of 6 miles in 1.5 hours. Amanda would catch up to him in the same time, which is 1.5 hours.
Q: What if Amanda starts jogging at a different speed?
A: If Amanda starts jogging at a different speed, the problem becomes more complex. We would need to recalculate the relative speed between Amanda and Ben, and use this information to determine the time it takes for Amanda to catch up to Ben. For example, if Amanda starts jogging at 8 miles per hour, the relative speed between her and Ben would be 4 miles per hour. Using the same equation as before, we can determine that it would take Amanda 1 hour to catch up to Ben.
Q: What if Ben and Amanda are walking/jogging on a circular path?
A: If Ben and Amanda are walking/jogging on a circular path, the problem becomes even more complex. We would need to take into account the circumference of the circular path, and use this information to determine the time it takes for Amanda to catch up to Ben. For example, if the circular path has a circumference of 10 miles, and Ben travels at 4 miles per hour, he would cover a distance of 6 miles in 1.5 hours. Amanda would need to cover the same distance, plus the distance she travels in time t, to catch up to Ben. We can use the same equation as before to determine the time it takes for Amanda to catch up to Ben.
Q: Can we use this problem to model real-world scenarios?
A: Yes, this problem can be used to model real-world scenarios. For example, if we are trying to determine the time it takes for a car to catch up to a truck on the highway, we can use the same mathematical concepts as in this problem. We would need to take into account the speed of the car and the truck, as well as the distance between them, to determine the time it takes for the car to catch up to the truck.
Q: What are some real-world applications of this problem?
A: There are many real-world applications of this problem. For example, in logistics and transportation, we need to determine the time it takes for trucks to catch up to each other on the highway. In sports, we need to determine the time it takes for athletes to catch up to each other during a race. In finance, we need to determine the time it takes for investments to catch up to each other in value.
Conclusion
In this article, we answered some frequently asked questions related to the problem of Ben and Amanda's speedy chase along a path. We explored different scenarios, such as Ben and Amanda starting at the same point, Amanda starting jogging at a different speed, and Ben and Amanda walking/jogging on a circular path. We also discussed some real-world applications of this problem, and how it can be used to model real-world scenarios.