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Will the Train Stop in Time?

Date: 09/21/1999 at 17:57:48
From: Student
Subject: Train/speed/brakes

A train is traveling at 80.0 m/s when the engineer sees a car stalled 
on the tracks 2000 m ahead. He immediately applies the brakes and the 
train starts to slow down at the rate of 1.50 m/s^2. Will the train 
stop in time? If so, how far short does it stop? If not, with what 
speed does the train hit the car, and how far past the car does the 
train go?

Date: 10/08/1999 at 13:24:58
From: Doctor TWE
Subject: Re: Train/speed/brakes

Hi -

There are four formulae in physics that can be useful in solving this 
type of problem. They are:

     (1) s   = .5(u + v)t
     (2) v   = u + at
     (3) s   = ut + .5a(t^2)
     (4) v^2 = u^2 + 2as

   where:  s = distance traveled
           u = initial velocity (speed)
           t = time
           a = acceleration
           v = final velocity (speed)

Note that the acceleration will be negative because the train is 
decelerating (slowing down).

Let's first look at what we have, and what we need to know. We know 
the initial velocity of the train, u = 80 m/s. We also know that the 
acceleration a = -1.5 m/s^2. We don't know the stopping distance, s, 
(we want to see if it is less than 2000 m); nor do we know the time it 
takes to stop, t. If the train comes to a complete stop, we know the 
final velocity, v = 0.

We can start by finding the stopping distance (s) and see whether it 
is more or less than 2000 m. Both equation (1) and equation (3) solve 
for s, but to use either one we need to first find the stopping time, 
t. To find the time, we need to look at the equations with t. The only 
equation other than (1) and (3) that involves t is equation (2). Can 
we use it? We know a, u and v (once the train stops, the final 
velocity is zero), so we can use this to find t. Can you rearrange the 
equation to solve for t? If not, write back and I'll explain how to 
rearrange it.

Once we have the stopping time t, we can find the stopping distance 
using either equation (1) or (3). Which one should we use? How about 
trying both and seeing if you get the same answer. If you do - great; 
it's probably right. If you get two different answers, check your 
arithmetic and double-check your steps in finding the time.

Now we know whether the train stopped in time. If it did, we can also 
compute how far short of the car it stopped. (Remember that s was the 
stopping distance of the train, not how far short of the car it 
stopped.) If not, you want to find the speed at the time of impact and 
how far past the car the train traveled.

If the train hits the car, we need to calculate the impact velocity v. 
Lets re-evaluate what we know in this case. We know the initial 
velocity u, the acceleration a, and the distance s. The time we 
calculated above was to come to a complete stop, so it is not relevant 
if the train actually hits the car. Equations (1), (2) and (4) involve 
v, but (1) and (2) require t to find v. So we have to use equation 
(4). Since the left side of equation (4) is v^2, remember to take the 
square root after computing the stuff on the right side. (We want to 
find v, not v^2.)

How far the train travels after hitting the car depends in part on 
the effect the car has on the rate of the train's deceleration (the 
car will further "slow down" the train when they hit). For the sake of 
the problem, let's assume that the car's effect is negligible. This is 
not too unrealistic - the train has a mass many times that of the car, 
and therefore the car will not significantly affect it. (Imagine a 
bowling ball hitting a ping pong ball.) We can find the total distance 
the train travels using the method above, then compute how far past 
the car that is.

To see another similar problem, check out the following problem in our 
Ask Dr. Math archives:   

I hope this helps. Write back if you have any more questions.

- Doctor TWE, The Math Forum   
Associated Topics:
High School Physics/Chemistry

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