# Determine stopping distance of a train - modified Euler method

• Alexanddros81
In summary, In this problem, the acceleration is a=-(7/4)+(t/16) m/s^2, where t is the time in seconds measured from when the brakes are applied. The train stops at 138.7 meters using the modified Euler method with Δt=4s.
Alexanddros81

## Homework Statement

12.81[/B] A train traveling at 20m/s is brought to an emergency stop. During braking,
the acceleration is a=-(7/4)+(t/16) m/s^2, where t is the time in seconds measured
from when the brakes were applied. (a) Integrate the acceleration from t=0 to
t=16s using Euler's method with Δt=2s. (b) Use the results of the integration to
determine the stopping distance of the train and compare you answer with 138.7m,
the value found analytically.

12.82 Solve Prob. 12.81 using the modified Euler method with Δt=4s

## The Attempt at a Solution

12.81 [/B]was handled in my previous post in this sub forum. I attach my attempt in 12.82.
I have run an online modified Euler calculator and my values are not correct.

Can you check this?

Alexanddros81 said:

## Homework Statement

12.81[/B] A train traveling at 20m/s is brought to an emergency stop. During braking,
the acceleration is a=-(7/4)+(t/16) m/s^2, where t is the time in seconds measured
from when the brakes were applied. (a) Integrate the acceleration from t=0 to
t=16s using Euler's method with Δt=2s. (b) Use the results of the integration to
determine the stopping distance of the train and compare you answer with 138.7m,
the value found analytically.

12.82 Solve Prob. 12.81 using the modified Euler method with Δt=4s

## The Attempt at a Solution

12.81 [/B]was handled in my previous post in this sub forum. I attach my attempt in 12.82.
I have run an online modified Euler calculator and my values are not correct.
View attachment 210017

Can you check this?

No. Type it all out and I will be happy to help. Also: you do not state what formulas you use, so it is impossible to say whether you are doing it correctly. (There are several versions of Euler-type methods---modified or otherwise---so you need to be explicit about what you mean.)

Last edited:
what do you want me to type out besides the formulas?

Alexanddros81 said:
what do you want me to type out besides the formulas?

Starting with the formulas would be nice; then we could go on from there. For instance, are you just using some form of Euler on the single ##v'(t) =a(t)## (then somehow getting ##x(t)## from ##v(t)##), or are you applying an Euler method to the 2-dimensional linear, non-homogeneous system
$$\frac{d}{dt}\pmatrix{x(t) \\ v(t)} = \pmatrix{v(t) \\ a(t)} = \pmatrix{0 & 1\\0 & 0} \pmatrix{x(t)\\v(t)} + \pmatrix{0 \\ a(t)} ?$$

Last edited:
So here are the formulas:

##\hat v_{i+1}=v_i+a_i Δt##
##\hat x_{i+1}=x_i+v_i Δt##

##v_{i+1}=v_i+\frac {a_i + \hat a_{i+1}} {2} Δt##

##x_{i+1}=x_i+\frac {v_i + \hat v_{i+1}} {2} Δt##

##a_i=f(v_i, x_i, t_i)##
##\hat a_{i+1}=f(\hat v_{i+1}, \hat x_{i+1}, t_{i+1})##

I have attached the theory.

Do you want to upload the book example?

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Alexanddros81 said:
So here are the formulas:

##\hat v_{i+1}=v_i+a_i Δt##
##\hat x_{i+1}=x_i+v_i Δt##

##v_{i+1}=v_i+\frac {a_i + \hat a_{i+1}} {2} Δt##

##x_{i+1}=x_i+\frac {v_i + \hat v_{i+1}} {2} Δt##

##a_i=f(v_i, x_i, t_i)##
##\hat a_{i+1}=f(\hat v_{i+1}, \hat x_{i+1}, t_{i+1})##

I have attached the theory.

Do you want to upload the book example?

No, do not upload anything more.

The equations you wrote above are not the best ones for YOUR problem, which has acceleration ##a = ## known function of ##t## alone. You should use the equations on page 61, but the ones you wrote above are from page 62. Actually, for the case of ##\ddot{x} = a(t)## the equations on pages 61 and 62 are mathematically equivalent, but why do more work than you need to do? Why make things more complicated than they need to be?

Anyway, if you type out (not photograph) the results of the first two or three time-steps I will be glad to review them.

Oh, and by the way: the answer to the question I asked in post #4 (but which you never answered) is YES, you are applying Euler's method to the system I wrote there.

## 1. What is the modified Euler method?

The modified Euler method is a numerical method used to approximate the solution to a differential equation. It is an improvement upon the regular Euler method and involves using a midpoint approximation in addition to the regular Euler approximation.

## 2. How does the modified Euler method calculate the stopping distance of a train?

The modified Euler method uses the laws of motion and the train's initial velocity and acceleration to determine the position of the train at each time interval. By calculating the position at various time intervals, the method can approximate the stopping distance of the train.

## 3. What factors affect the accuracy of the modified Euler method in determining stopping distance?

The accuracy of the modified Euler method in determining stopping distance can be affected by factors such as the time interval used, the initial velocity and acceleration of the train, and any external forces acting on the train.

## 4. Can the modified Euler method be used for any type of train?

Yes, the modified Euler method can be used for any type of train as long as the necessary initial conditions and equations of motion are known. However, the accuracy of the method may vary depending on the specific characteristics of the train.

## 5. How does the modified Euler method compare to other methods for determining stopping distance?

The modified Euler method is a relatively simple and efficient method for determining stopping distance. However, it may not be as accurate as other more complex methods, such as numerical integration or computer simulations, which take into account more factors and variables.

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