# Final Velocity

If we have an object that is theoretically set to accelerate constantly for an unknown amount of time (disregarding the issues that alone brings up), what formula would be used to calculate its velocity? I know we can determine final velocity for after t amount of time, but would velocity-addition be more appropriate?

I'm attempting to teach myself physics and while I know of velocity-addition and final velocity, I haven't found a place that indicates under cases to use them.

Does anyone have a site they personally know that is any good for physics instruction?

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What do you mean by velocity addition?

$$a = \frac{d^2 x}{dt^2} = \frac{dv}{dt}$$

treat as differentials

$$\int_{t'=0}^{t'=t}a dt' = \int_{v(0)=v_0}^{v(t)} dv$$

gives an equation for velocity based on an initial velocity

$$v(t) = v_0 + at$$

If you want to get rid of the time dependencies then solve for t

$$t = (v - v_0)/a$$

Then because you know that (from the mean value theorem if you have done it)

$$x = \vec{v}t = (1/2)(v_0 + v)t$$

Where we have already figured out t that.

$$x = (v+v_0)(v-v_0)/2a$$

So solve for velocity

$$v^2 = v_0^2 + 2ax$$

I don't think that relying on websites to learn physics is such a good idea. I would see what the library has in terms of books. I think that Randall Knight has a pretty good introductory physics book if your library has it.

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Oh, I see what you are asking. You are wondering whether if some acceleration is applied to an object for long enough, do you have to consider relativistic effects? I guess the short answer would be yes. The thing is though that as the particles velocity increases, its mass will increase too, and so whatever is forcing it will have a harder and harder time accelerating it. There will, in a way, be a limiting acceleration (and therefore a limiting velocity) that will depend on the force and the mass.

The safest thing to do with relativity is to use the equations that always work, which are conservation of energy, and conservation of momentum. For example, rather than F=ma, which is true only in general (i.e. non-relativistic cases) you should use F = dp/dt.

Do you have a specific example, or are you just wondering?

No real specific example. It's just, I read about these formulas and such, but haven't found a place that really discusses to what, where and why they're used.

Okay. If you haven't figured out by now, the velocity addition formulas relate to special relativity. My previous suggestion still stands of finding a published book, Knight even covers special relativity if you can find the big book with all volumes in one, rather than trusting internet Joe Schmo.

Okay. If you haven't figured out by now, the velocity addition formulas relate to special relativity. My previous suggestion still stands of finding a published book, Knight even covers special relativity if you can find the big book with all volumes in one, rather than trusting internet Joe Schmo.
Yup, I've been reading a lot about GR and SR recently. But I'll start perusing for books. Thanks :D

Dick