# (Air Resistance) Distance with respect to time

1. Apr 13, 2012

### BitterX

1. The problem statement, all variables and given/known data

a ball with mass $m$ is thrown up from the ground with a velocity $v_0$

the force due to Air resistance is: $F=-bv$
(there's also gravity)

find the velocity and distance with respect to time

3. The attempt at a solution

I think I got the velocity:
$v=e^{-\frac{b}{m}t}(\frac{m}{b}g+v_0)-\frac{m}{b}g$

the units are okay and so does t=0 (v=v0)

I have a problem with the distance, I integrate:

$x=\int^{t}_{0}vdt=\int^{t}_{0}(e^{-\frac{b}{m}t}(\frac{m}{b}g+v_0)-\frac{m}{b}g) = -e^{-\frac{b}{m}t}(\frac{m}{b}g+v_0)\frac{m}{b}-\frac{m}{b}gt$

so the units are okay but if t=0 $x=-(\frac{m}{b}g+v_0)\frac{m}{b}-\frac{m}{b}gt$

and it should be 0 (or so I think)
maybe the constant of integration should be the expression I got?
I'm really confused.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Apr 13, 2012

### collinsmark

That looks right to me, so far.
I'm sure as you know, there are two ways to evaluate an integral. The first way is to find the anti-derivative, and then tack on an arbitrary constant at the end. Then use your initial conditions to solve for the arbitrary constant.

The other way, which you have started here, is to evaluate the definite integral, and apply the initial conditions as your limits of integration. For example, if F(t) is the anti-derivative of f(t), then

$$F(b) - F(a) = \int_a^b f(t)dt$$

Since you've already started with the definite integral method, let's go with that.
Okay, that looks like the correct anti-derivative to me, but you forgot to evaluate it over the limits from 0 to t.

Plugging in t = t is easy enough, but notice that the antiderivative is not 0, when evaluated at t = 0. This needs to be subtracted off. Remember,

$$F(b) {\color{red} {- F(a)}} = \int_{\color{red}{a}}^b f(t)dt$$

Last edited: Apr 13, 2012
3. Apr 14, 2012

### BitterX

Wow, I don't know how I missed it.

I plugged t=0 only in -(mg/b)t (so it was 0) and totally ignored the first part :/

Thank you!