# Closest approach of two trains on perpendicular tracks

• stunner5000pt
In summary, the conversation discusses the problem of determining the closest point in time for two trains, one traveling south at 60 km/hr and the other heading west at 45 km/hr. The solution involves using the Pythagorean formula to find the distance between the two trains at any given time and finding the minimum of that function to determine the closest time.
stunner5000pt
Kinda confused on this one becuase of the time difference involved

A train leaves a station going south at 60 km/hr at 10:00. Another train heading due west reaches this station at 11:00. The latter train was traveling at 45km/hr. At what time are they the closest?

To start with the 45 train is 45km away from teh station and in the end the distnace is 60 km
iformulated something like 45(1-t) + 60t = Distance
but that doesn't yield the answer i need...
can someone help!

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Call the point at which the 60 km/hr train starts the "origin." You know the direction and distance of each train from the origin at any given time.

As shown in your figure, the vectors from the origin to each train form a right triangle, the hypotenuse of which is the distance between them.

If you know the length of the triangle's legs as a function of time, use the Pythagorean formula to find the length of the triangle's hypotenuse as a function of time. Then, find the minimum of that function.

- Warren

Based on the given information, we can use the distance formula d = rt to determine the distance of each train from the station at any given time. Let's use t as the time in hours since 10:00. For the train heading south, its distance from the station can be represented as d = 60t. For the train heading west, its distance from the station can be represented as d = 45(1-t). We can set these two equations equal to each other to find the time when they are closest to each other:

60t = 45(1-t)
60t = 45 - 45t
105t = 45
t = 45/105
t = 0.43 hours

Therefore, at 10:26 (10:00 + 0.43 hours), the trains will be closest to each other. This is because at this time, the southbound train will be 25.8 km away from the station, while the westbound train will be 19.4 km away. This is the minimum distance that they will be from each other during their journey. I hope this helps clarify the situation.

## What is meant by "closest approach of two trains on perpendicular tracks"?

The closest approach of two trains on perpendicular tracks refers to the point at which the two trains are the closest to each other as they pass on tracks that are at a 90-degree angle to each other. This occurs when the trains are approaching each other from opposite directions on these perpendicular tracks.

## How is the closest approach of two trains on perpendicular tracks calculated?

The closest approach can be calculated by using the distance formula for two points. The two points in this case would be the center of each train as they pass each other. The formula is: d = √[(x2-x1)^2 + (y2-y1)^2], where d is the distance between the two points, x1 and y1 are the coordinates of the first point, and x2 and y2 are the coordinates of the second point.

## What factors affect the closest approach of two trains on perpendicular tracks?

The closest approach of two trains on perpendicular tracks can be affected by several factors, including the speed of the trains, the distance between the tracks, the angle at which the tracks intersect, and the acceleration or deceleration of the trains as they approach each other.

## Why is it important to calculate the closest approach of two trains on perpendicular tracks?

Calculating the closest approach of two trains on perpendicular tracks is important for safety reasons. It allows train operators to determine the minimum distance between the trains and take necessary precautions to avoid collisions. It also helps with train scheduling and efficiency, as knowing the closest approach can prevent delays and improve overall train operations.

## What other applications does the concept of closest approach have?

The concept of closest approach has many other applications in science and engineering, such as in celestial mechanics to calculate the closest distance between planets or other celestial bodies. It is also used in collision avoidance systems for vehicles, drones, and satellites. The concept can also be applied in mathematics and physics to calculate the minimum distance between two objects moving in a specific way.

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