# Homework Help: Physics dynamic problem

1. May 26, 2007

### hastings

1. The problem statement, all variables and given/known data
An object with mass m=0.1 kg is thrown with an initial speed v0=20m/s in a viscous matter which opposes its motus with a resistance force of F=-Bv (v is the speed and B=2kg/s). Find the distance this object has covered in the viscous matter. Leave aside gravity force.

2. Relevant equations
F=m*a

3. The attempt at a solution
I tried this way

$$\vec F=-\beta \vec v=m \cdot a$$

$$\frac{-\beta \vec v}{m}= a= \frac{dv}{dt}=\frac{ds}{d^2t}$$

$$\frac{-\beta \vec v}{m} d^2t=ds$$

$$\int{\frac{-\beta \vec v}{m} }d^2t=\int ds$$

$$\int{\frac{-\beta \vec v}{m} } \cdot t dt=s$$

$$\displaystyle{-\frac{1}{2}} \frac{\beta v }{m}t^2+K=s$$

How to continue? Is it right till this point?

2. May 26, 2007

### Staff: Mentor

Realize that v is not a constant but a function of t. Stick with this:
$$\frac{-\beta \vec v}{m}= a= \frac{dv}{dt}$$

Rearrange so that v is on one side and t on the other. Then you can integrate.

3. May 26, 2007

### hastings

you mean like this?
$$-\frac{\beta dt}{m}=\frac{1}{v}dv$$

$$\int{ -\frac{\beta}{m}}dt=\int \frac{1}{v}dv$$

$$-\frac{\beta}{m}t=ln(v)+K$$

How to get K? Should i substitute t0=0s & v0=20m/s?

4. May 26, 2007

### hastings

to get the distance "s", I need a formula with a, t and v
Where do I go with an expression with 2 unknown quantities, v and t?
$$-\frac{\beta}{m}t=ln(v)-ln20$$

5. May 26, 2007

### hastings

suppose I do ...
$$-\frac{\beta}{m}t+\log20=\log v$$

$$\frac{ds}{dt}=v=e^{-\frac{\beta}{m}t+\log20}$$

$$\int{ds}=\int{e^{-\frac{\beta}{m}t+\log20} }dt$$

$$\int ds=\int (C \cdot e^{\alpha t})dt \ \ \mbox{where \alpha=\frac{-\beta}{m} } \ \mbox{ and C=e^{\log20} }$$

$$s=\frac{C}{\alpha}e^{\alpha t}+K_1$$

I assume k1=0; I know all the quantities except t

Last edited: May 26, 2007
6. May 26, 2007

### Staff: Mentor

Looks good to me (with a bit of correction).

You can simplify a bit. For example:
$$C=e^{\ln20}$$
What does that equal?

Realize that k1 cannot equal zero if s = 0 at t = 0.

Last edited: May 26, 2007
7. May 27, 2007

### hastings

ok, Doc Al. got your message about posting the same topic more than once.
Won't do it again. This is what happened: after waiting a long time I didn't get much help in this section so I tried my luck on some other section - where eventually I got some interesting hint to solve this problem (THANKS A LOT TO "arunbg"!!).
Here's what I did

$$F_R= -\beta \cdot v=m \cdot a$$

$$(1) \ a=-\frac{\beta v}{m}$$

$$(2) \ a=v \frac{dv}{dx} \ \mbox{ infact } \displaystyle{(v \cdot \frac{ \frac{dv}{dt}}{\frac{dx}{dt} } = v \cdot \frac{a}{v} = a)}$$

$$v \frac{dv}{dx}= -\frac{\beta v}{m}$$

$$\frac{dv}{dx}= -\frac{\beta}{m}$$

$$dv=-\frac{\beta}{m} dx$$

$$\int{dv}= \ -\int{\frac{\beta}{m}dx}$$

$$v=-\frac{\beta}{m}x$$

$$x=-\frac{mv}{\beta}$$

$$x=-\frac{0.1 \mbox{ kg } (20 \mbox{ m/s})}{2 \mbox{ kg/s}}=-1m$$

I'm not sure whether this is the correct way, but the result is 1m.

8. May 27, 2007

### hastings

$$C=e^{\ln 20} \ \Longrightarrow C=20?$$

You are right about k1. with t=0 and x=0 i get

$$0=\frac{20}{\alpha} e^{\alpha \cdot 0}+K_1$$

$$0=\frac{20}{\alpha}+K_1 \ \Longrightarrow K_1=-\frac{20}{\alpha}$$

9. May 27, 2007

### hastings

Suppose I try with the definite integral...

Last edited: May 27, 2007
10. May 27, 2007

### hastings

Suppose I try with the definite integral...

$$a=\frac{dv}{dt}=-\frac{\beta v}{m}$$

$$\frac{1}{v}dv=-\frac{\beta}{m}dt$$

$$\int_{v_0}^v{\frac{1}{v}}dv=-\frac{\beta}{m} \int_0^t{}dt$$

$$\ln v -\ln v_0=-\frac{\beta}{m}t$$

$$\ln \frac{v}{v_0}=-\frac{\beta}{m}t$$

$$\frac{v}{v_0}=e^{-\frac{\beta}{m}t}$$

$$v=\frac{dx}{dt}=v_0e^{-\frac{\beta}{m}t}$$

$$dx=v_0e^{-\frac{\beta}{m}t}dt$$

$$\int_0^x{dx}= \int_0^t {v_0e^{-\frac{\beta}{m}t}}dt$$

$$x=v_0[-\frac{m}{\beta}e^{-\frac{\beta}{m}t}]_0^t=v_0[\frac{m}{\beta}-\frac{m}{\beta}e^{-\frac{\beta}{m}t}]= \frac{m v_0}{\beta}[1-e^{-\frac{\beta}{m}t}]$$

So in the end I get

$$x=\frac{mv_0}{\beta} \displaystyle{[1-e^{-\beta t/m }]}$$

How to go foward now? The answer is 1m

Last edited: May 27, 2007
11. May 27, 2007

### hastings

What I don't know is the instant (time) this object stops or anyway comes out of the viscous medium. Just a step away from the solution...How do I go on now?

Anybody there to help me? Any suggestion is appreciated.

Last edited: May 27, 2007
12. May 27, 2007

### cristo

Staff Emeritus
This method is fine up until you integrate and decide to omit the constant of integration!! The next line should read

$$v=-\frac{\beta}{m}x +A$$ for some constant, A. Use your initial conditions to find A, then proceed as you did to find x.

13. May 27, 2007

### hastings

Assuming that "t" is endless (of course not negative, never heard of "t=-4s")

$$\lim_{t \to +\infty}\frac{mv_0}{\beta} \displaystyle{[1-e^{-\beta t/m }]}= \frac{mv_0}{\beta}$$ $$= \frac{0.1kg \ (20 \mbox{ m/s})}{2kg/s}=1m$$ :rofl:

Last edited: May 27, 2007
14. May 27, 2007

### Staff: Mentor

You seem to be doing the entire problem over and over several ways. Nothing wrong with that as long as you realize that any correct way will give you the answer if you stick with it. You had all you needed for the answer in post #5, once you found the correct constant.
Examining this solution should tell you something about how the object will move. The distance reaches a limit as time goes to infinity.

Take the limit.

There you go.

Just for fun, if you can stand it, plug in a few values of time (1 sec, 2 sec, etc...) to see that you don't really have to wait forever for the object to go pretty much as far as it will go.