# Effect of air resistance

## Homework Statement

The effect of air resistance is to slow down an object. It can be shown that the height of a falling object is given by the following:

y=$$y_{o}$$ - [t + ($$e^{-bt}$$ - 1) / b] * g/b.

Show that for short times the eqn is reduced to

y=$$y_{o}$$ - 0.5(g)(t^2)

## Homework Equations

I think the regular distance formula yf = yi(t) + 0.5(at^2) is what i need here because the reduced eqn resembles it very cloesely.

## The Attempt at a Solution

I've tried to do a lim as t approaces 0 on the first eqn. I've tried to equate the second eqn with the first eqn. I've been at it for a couple of hours now.

ANY help would be appreciated.

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rock.freak667
Homework Helper
Try the Taylor expansion of exp(-bt) and ignore terms past t2

ideasrule
Homework Helper
The distance formula yf = yi(t) + 0.5(at^2) is actually the same as y=y0 - 0.5(g)(t^2), the equation the question wants you to prove, since a=-g.

Do you know what the Taylor series expansion of e^x is? If you do, expand e^-bt, discard higher-ordered terms, and y will reduce to y0 - 0.5(g)(t^2).

ideasrule
Homework Helper
Oops, I posted my answer before I saw rock.freak's.

Hi,
thankyou very much. I have gotten the reduced equation.

but can someone explain to me why the taylor expansion was needed? what does the expansion have to do with "time being very short"???
also, why are only the terms until t^2 needed?

thanks

rock.freak667
Homework Helper
Hi,
thankyou very much. I have gotten the reduced equation.

but can someone explain to me why the taylor expansion was needed? what does the expansion have to do with "time being very short"???
also, why are only the terms until t^2 needed?

thanks
if t is small, terms like t3,t4 and higher will give even smaller numbers. So depending on the degree of accuracy, these numbers don't affect the desired accuracy.

so if i have understoon taylor series correctly, it just means the

function e^x (or in my case e^-bt) can be REWRITTEN as a sum of individual terms given by the taylors series. Am i correct?

and if I am correct then in my question, the higher the degree on t, the smaller the number.

so my third question is why stop at t^2?

ideasrule
Homework Helper
"function e^x (or in my case e^-bt) can be REWRITTEN as a sum of individual terms given by the taylors series. Am i correct?"

Yup. The more terms you include, the more accurate the approximation. The Taylor series expansion becomes infinitely accurate with an infinite number of terms.

"and if I am correct then in my question, the higher the degree on t, the smaller the number."

Exactly.

"so my third question is why stop at t^2?"

Because that's the level of approximation that gives you y=y0 - 0.5(g)(t^2). If you include more terms, you'll get a more accurate equation, but it won't be the same as the free-fall equation.