[R] newbie's additional (probably to some extent OT) questions

[JB]

(1)
So finally, thank to your help I have this:

  summary(lm(x ~ 0+I(t^2)))

And then I get this result:
=================================================
Call:
lm(formula = x ~ 0 + I(t^2))

Residuals:
        Min         1Q     Median         3Q        Max
-3.332e-02 -9.362e-03  1.169e-05  1.411e-02  3.459e-02

Coefficients:
         Estimate Std. Error t value Pr(>|t|)
I(t^2) 0.0393821  0.0001487   264.8   <2e-16 ***
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

Residual standard error: 0.01945 on 18 degrees of freedom
Multiple R-Squared: 0.9997,     Adjusted R-squared: 0.9997
F-statistic: 7.014e+04 on 1 and 18 DF,  p-value: < 2.2e-16
=================================================

I see in MuPad, that Delta^2 is 0.006813. Now is not the standard error the 
square root of Delta^2? Should I not get 0.069 as standard error?

(2)
When I use the model
summary(lm(x ~ I(t^2)))
I get (of course) another result with a slightly smaller Delta^2. But I do 
not expect such an error as this would mean that there was a systematic 
error in our measurement of the distance and if I understand the result of 
R correctly, the error was 0.04m which is impossible:

==========================================================
Call:
lm(formula = x ~ I(t^2))

Residuals:
        Min         1Q     Median         3Q        Max
-0.0202520 -0.0116533 -0.0006036  0.0036699  0.0432987

Coefficients:
              Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0427606  0.0161085   2.655   0.0167 *
I(t^2)      0.0379989  0.0005367  70.801   <2e-16 ***
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

Residual standard error: 0.01683 on 17 degrees of freedom
Multiple R-Squared: 0.9966,     Adjusted R-squared: 0.9964
F-statistic:  5013 on 1 and 17 DF,  p-value: < 2.2e-16
=====================================================

What is going on here?
(Sorry but I am only a high school teacher and have not much idea of 
statistics.)

TIA,

JB