Falling object and air resistance

[justin016]

Homework Statement

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

Homework Equations

y=y0-[t+(e^-bt - 1)/b]g/b

a. show that for short times this reduces to the expected expression
y=y0-1/2gt^2

b. Find the velocity
c. Find the acceleration

The Attempt at a Solution

I really not sure where to start with part a. what does it mean by reduces?

 

[Delphi51]

You must replace the exponential with its power series. The power series will be a constant term plus a term with t, plus a t^2 term, plus a t^3 term and so on. For small times you can ignore the higher power terms. In this case keep only the first 3 terms.
If you aren't familiar with power series, you can look them up in the back of the book or in a reference book with tables of integrals. The exponential one is also given here:
http://en.wikipedia.org/wiki/Power_series

 

[KL90]

You must replace the exponential with its power series. The power series will be a constant term plus a term with t, plus a t^2 term, plus a t^3 term and so on. For small times you can ignore the higher power terms. In this case keep only the first 3 terms.
If you aren't familiar with power series, you can look them up in the back of the book or in a reference book with tables of integrals. The exponential one is also given here:
http://en.wikipedia.org/wiki/Power_series

Could you help explain how to find the velocity and the acceleration for the above equation. I know you differentiate it once for the velocity and then differentiate the velocity for the acceleration, but I'm having trouble.

 

[Delphi51]

The expression for y is pretty ugly; I would expand it out like this:

Go ahead and try to find dy/dt. I'm sure you can do the constant terms and the one with a t factor. For the exponential, recall that the derivative of e^x is e^x.