Determine stopping distance of a train - modified Euler method

[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

Homework Equations

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?

 

[Ray Vickson]

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

Homework Equations

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.)

 

[Alexanddros81]

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

 

[Ray Vickson]

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
$$\frac{d}{dt}\pmatrix{x(t) \\ v(t)} = \pmatrix{v(t) \\ a(t)} = \pmatrix{0 & 1\\0 & 0} \pmatrix{x(t)\\v(t)} + \pmatrix{0 \\ a(t)} ? $$

 

[Alexanddros81]

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?

 

[Ray Vickson]

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.