# A two trains problem (distance and speed)

• missrikku
In summary, the problem involves two trains with equal speeds and a bird flying at a different speed between them. The bird flies back and forth between the trains until they collide. Using the given information, the total distance the bird travels can be calculated by finding the time it takes for the trains to collide and using the bird's speed to determine the total distance it travels. The answer is 6.0 x 10^4 m. The process used to solve this problem was correct, although there is a more difficult way to solve it by summing an infinite series.
missrikku
Hello, this is the problem:

Two trains, each having a speed of 30 km/hr, are headed at each other on the same straight track. A bird that can fly 60 km/hr flies off the front of one train when they are 60 km apart and heads directly back for the other train. On reaching the other train it flies directly back to the first train, and so forth. What is the total distance the bird travels?

With this information, I came up with the following:

Vbird = 60 km/hr = 16.667 m/s
Vtrain1 = 30 km/hr = Vtrain2 = 8.333 m/s
d = distance between trains when bird 1st flies off = 60km

I) I found out how long it would take for the cars to hit each other:

Vtrain1 = Vtrain2 = 8.333 m/s = 3.0 x 10^4 m / t
** 3.0 x 10^4 m came from doing d/2 or 60km/2 = 30km
t = 3600s

II) I used Vbird to find the total distance it traveled:

16.667 = total distance / 3600s
total distance = 6.0 x 10^4 m

Giving me the ans: 6.0 x 10^4 m

Was my process for answering this problem correct?

Yes, you know the speed of the bird and you know time it was flying: the product gives the total distance flown.

Of course you could try to do it the HARD way: calculate each leg back and forth and do it as an infinite series!

There is an old story about a famous mathematician (VonNeumann? I've also heard it about Wiener.) that a man asked this problem of him and he thought for a few seconds and immediately gave the correct answer. The man chuckled and said "A lot of people try to do that by summing the infinite series." VonNeumann looked puzzled and said "But I did sum an infinite series!"

(You are now to roll on the floor laughing!)

scottdave

Yes, your process for answering this problem is correct. You correctly identified the variables and used the given information to solve for the total distance the bird traveled. Your calculation for the time it takes for the trains to hit each other is also correct. Overall, your solution is clear and accurate. Good job!

## 1. What is a two trains problem?

A two trains problem is a type of math problem that involves two trains traveling towards each other, with different speeds and distances. The goal is to determine how long it will take for the two trains to meet, and at what distance from their starting points.

## 2. How do you solve a two trains problem?

To solve a two trains problem, you need to first identify the given information, such as the speeds and distances of the trains. Then, you can use the formula: time = distance / speed to calculate how long it will take for each train to reach their meeting point. Finally, you can add these times together to get the total time it will take for the two trains to meet.

## 3. What if the trains are not traveling towards each other?

If the trains are not traveling towards each other, but in the same direction, the formula to use is: time = distance / (speed1 - speed2). This is because the relative speed between the two trains is the difference between their speeds when traveling in the same direction.

## 4. Can you use the same approach for trains with multiple stops?

The approach for solving a two trains problem can be applied to trains with multiple stops, but it may be more complicated. In this case, you would need to consider the distance and speed for each segment of the journey, and then add up the times for each segment to get the total time for the entire journey.

## 5. What are some real-life applications of a two trains problem?

A two trains problem can be applied to real-life scenarios, such as calculating the arrival time of two trains at a station, determining the distance between two cities based on the train schedules, or even in traffic engineering to optimize train speeds and schedules.

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