Finding velocity vs. time, when acceleration is dependent on velocity

[KingNothing]

Hello all. I have been struggling with this for a bit. I will present it as generally as I can.

Say I have an object with an initial velocity v0. It is only acted upon by an acceleration function, a(v), which is a function of v. How can I find a function for v(t)? Does this require a differential equation?

 

[Mute]

By definition,

$$a = \frac{dv}{dt}$$

Since you know a(v), you can rearrange and integrate:

$$dt = \frac{dv}{a(v)}$$

This will give you t as a function of v (if you can do the integral), which you can in principle invert to get v(t).

 

[KingNothing]

By definition,

$$a = \frac{dv}{dt}$$

Since you know a(v), you can rearrange and integrate:

$$dt = \frac{dv}{a(v)}$$

This will give you t as a function of v (if you can do the integral), which you can in principle invert to get v(t).

Okay. Thanks for the start. Say for example I have a(v) = 5v, and initial velocity v0 = 10 m/s.

$$dt = \frac{dv}{a(v)}$$
$$t = \int \! \frac{1}{a(v)} \, \mathrm{d}v = \! \frac{ln(v)}{5}$$
$$v = e^{5t}$$

This looks great so far; how do I incorporate an initial value?

 

[pwsnafu]

This looks great so far; how do I incorporate an initial value?

You need a constant of integration when doing the integrals.

 

[KingNothing]

You need a constant of integration when doing the integrals.

$$t = \int \! \frac{1}{a(v)} \, \mathrm{d}v = \! \frac{ln(v)}{5} + C$$
$$v = e^{5t-5C}$$

I'm still not totally sure how to incorporate it. It doesn't really make dimensional sense to just plug it in for C.

 

[Mute]

$$t = \int \! \frac{1}{a(v)} \, \mathrm{d}v = \! \frac{ln(v)}{5} + C$$
$$v = e^{5t-5C}$$

I'm still not totally sure how to incorporate it. It doesn't really make dimensional sense to just plug it in for C.

You know that at t=0, v=v_0. So, plug these into your formula for v and then solve for C. Or, in this case you just find that $v_0 = e^{-5C}$, so you can just replace the entire $e^{-5C}$ with v_0 to get $v(t) = v_0e^{5t}$.

 

[KingNothing]

Thank you guys so much! I get it now. Just an FYI, I was asking because I'm trying to use the drag equation, which when applied to an object of constant mass, is one such example of a velocity-dependent acceleration!