# Thread: First Order Error Analysis for O.D.E.

1. ## First Order Error Analysis

Given that the velocity of the falling parachutist can be computed by,

$v(t) = \frac{gm}{c}(1-e^{(\frac{c}{m})t})$

Use a first-order error analysis to estimate the error of v and t = 6, if g = 9.8 and m = 50 but c = 12.5 + or - 1.5.

I'm having a tough time starting this one. Can someone nudge me in the right direction?

EDIT: I just realized I posted this is the wrong section, it's clearly not an ODE can someone please move it?

2. Originally Posted by jegues
Given that the velocity of the falling parachutist can be computed by,

$v(t) = \frac{gm}{c}(1-e^{(\frac{c}{m})t})$

Use a first-order error analysis to estimate the error of v and t = 6, if g = 9.8 and m = 50 but c = 12.5 + or - 1.5.

I'm having a tough time starting this one. Can someone nudge me in the right direction?

EDIT: I just realized I posted this is the wrong section, it's clearly not an ODE can someone please move it?
Click the traingle with ! in the middle and type in the forum it should be in and hit submit.

3. Originally Posted by jegues
Given that the velocity of the falling parachutist can be computed by,

$v(t) = \frac{gm}{c}(1-e^{(\frac{c}{m})t})$

Use a first-order error analysis to estimate the error of v and t = 6, if g = 9.8 and m = 50 but c = 12.5 + or - 1.5.

I'm having a tough time starting this one. Can someone nudge me in the right direction?

EDIT: I just realized I posted this is the wrong section, it's clearly not an ODE can someone please move it?
Start by expanding as a Taylor series (in c) about c=12.5 and truncate after the first term.

CB

4. Originally Posted by CaptainBlack
Start by expanding as a Taylor series (in c) about c=12.5 and truncate after the first term.

CB
So,

$v(t) = f(t) + R_{n}$

$v(6) = g(50)(1-e^{\frac{12.5 \times 6}{50}}) + R_{n} = -1706.03 + R_{n}$

How do I get the error from this?

5. Originally Posted by jegues
So,

$v(t) = f(t) + R_{n}$

$v(6) = g(50)(1-e^{\frac{12.5 \times 6}{50}}) + R_{n} = -1706.03 + R_{n}$

How do I get the error from this?
After the first non-constant term.

CB

6. Originally Posted by CaptainBlack
After the first non-constant term.

CB
Here's what I came up with.

I'm pretty sure I must be misunderstanding something because I dont see any nonconstant terms.

We know the values for every variable so we should just end up with a number, no?

Where did I go wrong?

EDIT: Don't worry about the -1706.03 in the first line, it's not supposed to be there.

7. Originally Posted by jegues
Given that the velocity of the falling parachutist can be computed by,

$v(t) = \frac{gm}{c}(1-e^{(\frac{c}{m})t})$

Use a first-order error analysis to estimate the error of v and t = 6, if g = 9.8 and m = 50 but c = 12.5 + or - 1.5.

I'm having a tough time starting this one. Can someone nudge me in the right direction?

EDIT: I just realized I posted this is the wrong section, it's clearly not an ODE can someone please move it?
$v(t,c=12.5+\varepsilon)=\dfrac{gm}{12.5+\varepsilo n}(1+e^{(12.5+\varepsilon)t/m})$

Now expand the right hand side as a series in $\varepsilon$

CB

8. Originally Posted by CaptainBlack
$v(t,c=12.5+\varepsilon)=\dfrac{gm}{12.5+\varepsilo n}(1+e^{(12.5+\varepsilon)t/m})$

Now expand the right hand side as a series in $\varepsilon$

CB
Okay so,

$v(t) = \dfrac{gm}{12.5+\varepsilon}(1+e^{(12.5+\varepsilo n)t/m}) + \dfrac{gm}{12.5+\varepsilon}(1+\frac{(12.5+\vareps ilon)}{m}e^{(12.5+\varepsilon)t/m}) + ... + R_{n}$

What's next?

9. Originally Posted by jegues
Okay so,

$v(t) = \dfrac{gm}{12.5+\varepsilon}(1+e^{(12.5+\varepsilo n)t/m}) + \dfrac{gm}{12.5+\varepsilon}(1+\frac{(12.5+\vareps ilon)}{m}e^{(12.5+\varepsilon)t/m}) + ... + R_{n}$

What's next?
Is that a power series in $\varepsilon$?

CB

10. Originally Posted by CaptainBlack
Is that a power series in $\varepsilon$?

CB
I'm confused as to what you want me to do.

Do you mean this,

$e^{12.5 + \varepsilon } = \sum \frac{(12.5 + \varepsilon)^{n}}{n!}$

The sum starts from n=0 and goes to infinity, I can't seem to put it in.