# Train collision (linear kinematics)

• walking
In summary: So, in summary, the minimum deceleration needed to avoid a collision between the passenger train and the freight train is -0.703 m/s^2. This can be found by setting up a system of equations and solving for the acceleration. However, it is important to note that the passenger train does not have to come to a complete stop at the time of collision, but rather just needs to have a velocity of 0 relative to the freight train. Another approach is to do all calculations in the rest frame of the freight train, which would result in the same minimum deceleration of -0.703 m/s^2.
walking

## Homework Statement

A passenger train is traveling at 29ms^-1 and a freight train is traveling 360m ahead of it at 6ms^-1 in the same direction on the same track. The driver of the passenger train has a reaction time of 0.4s before he starts decelerating. What is the minimum deceleration to avoid colliding with the freight train?

x-0=29t+0.5at^2
0=29+at
x-350.8=6t

## The Attempt at a Solution

I tried solving this as follows but my answer is wrong according to a solution I found online. Where am I going wrong?

First I calculated that when the driver starts decelerating, the separation of the trains will be 350.8m. I then let the initial time be this point in time (ie when they are 350.8 apart) and I let x=0 at t=0. So initially, the passenger train is at x=0 and the freight train is at x=350.8. I let the time they meet be t and the displacement of both trains be x at this time. Also, the minimum deceleration would be if the passenger train stops exactly when they meet.

Based on this, I got the following equations:

For the passenger train we get two equations:
x-0=29t+0.5at^2
0=29+at (since it comes to a stop when they meet)

For the freight train:
x-350.8=6t

Solving these, I got a=-0.703, however the solution I found online (in a pdf document of solutions to the problems in Tipler/Mosca 5th edition; this problem is chapter 2 q96) gives -0.754.

walking said:
Also, the minimum deceleration would be if the passenger train stops exactly when they meet.
Why would the passenger train have to stop at that point?

Doc Al said:
Why would the passenger train have to stop at that point?
I thought it would give the slowest rate of deceleration if it comes to a stop at the maximum displacement to avoid collision, ie right when they meet. Maybe my intuition is wrong?

walking said:
I thought it would give the slowest rate of deceleration if it comes to a stop at the maximum displacement to avoid collision, ie right when they meet. Maybe my intuition is wrong?
Yes, your intuition is wrong. What you want to ensure is that when they meet the passenger train doesn't overtake the freight train (and thus collide). But to do that, it doesn't have to stop dead. Hint: The freight train is moving.

walking
Another suggestion: Do everything in the rest frame of the freight train instead.

Orodruin said:
Another suggestion: Do everything in the rest frame of the freight train instead.
In which case the OP's intuition would indeed be correct - final velocity of train relative to freight train would be zero!

neilparker62 said:
In which case the OP's intuition would indeed be correct - final velocity of train relative to freight train would be zero!
Indeed.

## What is a train collision?

A train collision is an accident where two or more trains collide with each other on the same track or at a crossing.

## What causes train collisions?

Train collisions can be caused by a variety of factors, including human error, mechanical failure, track defects, and signal malfunctions.

## How can train collisions be prevented?

Train collisions can be prevented through strict adherence to safety protocols, regular maintenance and inspection of trains and tracks, and the use of advanced technology such as automatic braking systems.

## What are the potential consequences of a train collision?

The consequences of a train collision can include injuries or fatalities to passengers and crew, damage to property and infrastructure, and disruption to train services.

## What is linear kinematics and how does it relate to train collisions?

Linear kinematics is a branch of physics that studies the motion of objects along a straight line. It is relevant to train collisions as it helps to understand the velocity, acceleration, and forces involved in a train collision and how they contribute to the severity of the accident.

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