Finding velocity function from a model of force

[Falken_47]

Homework Statement

Hi, I've got an assignment which involves finding the function of velocity against time, given the information that the particle is moving with intial velocity of v₀ and that it experience a resistive force of -βv²

Homework Equations

From the problem I've derived model of: ma = -βv²

The Attempt at a Solution

re-arranging the model, I have equation of:

dv/dt = -β/m * v²

then, using techniques from seperable equation,

dv/v² = -β/m * dt

-1/v = -β/m * t + C₁

pluging in v(0) = v₀

-1/v₀ = -β/m * 0 + C₁
Or
C₁ = -1/v₀

Now, plugging in C and rearraging the equation, I obtained

-1/v = -β/m * t - 1/v₀

v = m/(β * t) + v₀

So as you can see, I'm able to derived the velocity function, but the problem lies to the fact that if I try to plug in t=0 for the velocity, then the function will not work as division by zero is not allowed

Therefore, I'm thinking that something is wrong with my velocity function but up till now I can't still figure it out

Thank you in advance!

 

[cepheid]

Welcome to PF!

Your calculus is fine, you just forgot how to do division:$$\frac{1}{v(t)} = \frac{\beta}{m} t + \frac{1}{v_0}$$$$\Rightarrow v(t) = \left[ \frac{\beta}{m} t +\frac{1}{v_0} \right]^{-1} \neq \frac{m}{\beta t} + v_0$$

 

[Falken_47]

Ah yes thank you very much at pointing out my mistake, guess I spend too much time thinking how to actually solve the differential equation that I forgot about that. In any case, thank you for helping me again :)