# Two decelerating trains on a collision course

• marksyncm
In summary, the red train had a velocity of 0m/s and the green train had a velocity of 10m/s at the time of collision.
marksyncm

## Homework Statement

A red and a green train are headed towards each other on a collision course.

Red train velocity = 20m/s
Green train velocity = 40m/s

When the trains are a distance of 950 meters apart, they begin to decelerate at a steady pace of 1m/s^2.

1) Will the trains collide?
2) If so, what will be their respective speeds at the time of the collision?

## Homework Equations

##v=v_o+at## and ##x-x_0 = v_0t + \frac{at^2}{2}##

## The Attempt at a Solution

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Actually, I was able to obtain a solution (the trains collide; the red train has a velocity of 0m/s and the green train a velocity of 10m/s), but I am not sure if there is an easier way to do it. Here are the steps I took (without equations):

# Check if the trains collided:

1) Calculate total distance traveled by each train before it comes to a halt.
2) Is combined distance traveled by both trains equal to or greater than 950 meters? Yes - then the trains collided.

# Calculate the velocity of each train at collision time (this is where I'm most uncertain)

First, I decided to determine if the slower of the two trains (the red one) came to a complete stop prior to the collision. To do this:

1) Calculate the time ##t## it takes the red train to come to a stop.
2) Check the distance ##d_1## traveled by the red train in the above time.
3) Check the distance ##d_2## traveled by the green train in time ##t##.
4) Is ##d_1+d_2 < 950##? If so, the trains collided when the red train had a velocity of 0 m/s and the green train had whatever velocity it has when it traverses a distance of ##950-d_1##.
5) If ##d_1+d_2 > 950## , then both trains had a velocity ##>0## at the time of collision. These velocities can be calculated by determining the time of impact using the relativity of motion - in this case, ##60t - t^2 = 950## - then checking the velocity of each train at that time.

I'm wondering if there's a simpler approach that works regardless of whether both or only one of the two trains were moving at the time of impact? It seems that when I try the "relativity of motion" approach for cases where one of the trains came to a stop before the collision, I get a solution that's a complex number, and the real part of that number is actually a correct solution. It seems intuitive to me that the relativity of motion approach should not work in these cases (because the combined deceleration of the system changes after one of the trains comes to a stop), but I can't shake this feeling that there's a simpler approach to these problems.

PeroK
I can't see a problem with your approach. It's a good point that the motion is potentially in two phases: both trains decelerate for a time, then only one decelerates.

marksyncm
Thank you.

## What is the concept of two decelerating trains on a collision course?

The concept of two decelerating trains on a collision course refers to a scenario where two trains are moving towards each other and are both slowing down, with the intent to stop before they collide.

## What factors determine the outcome of a collision between two decelerating trains?

The outcome of a collision between two decelerating trains depends on several factors, including the initial speeds of the trains, the rates at which they are decelerating, and the distance between them at the start of the deceleration.

## How does the stopping distance of a train affect the likelihood of a collision?

The stopping distance of a train is a crucial factor in determining the likelihood of a collision. A longer stopping distance means the train needs more time to come to a complete stop, which increases the risk of a collision if the other train is also slowing down.

## Can the outcome of a collision between two decelerating trains be predicted?

Yes, the outcome of a collision between two decelerating trains can be predicted using mathematical equations and physics principles. However, it is important to note that external factors such as weather conditions or mechanical failures can also impact the outcome.

## Are there any safety measures that can be taken to prevent a collision between two decelerating trains?

Yes, there are safety measures that can be taken to prevent a collision between two decelerating trains. These include maintaining proper communication between train operators, implementing speed limits and signals, and using automatic braking systems.

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