Instantaneous velocity when non-constant acceleration

[mireazma]

Hello.
I've seen on Wikipedia (http://en.wikipedia.org/wiki/Airsoft_pellets#pellet_ballistics) a formula to find the aerodynamic drag (velocity dependent negative acceleration) for a spherical body. Knowing body's mass, diameter and air density I simplified it to the form:
A = v_t² * 0.04
where A is the drag, t is a point in time, v_t is the instantaneous velocity at t.
The problem comes now: how can I get v_t at arbitrary times or distances? A changes in instants of time so I can't do a simple
v_t = v₀ + A * t
Say the ball has an initial v₀ = 120m/s; what's the velocity at t = 1s?

 

[ShayanJ]

You should write Newton's second law: $\vec F_{net}=m \vec a$ where $\vec F_{net}$ is the vector sum of all forces on the mass m. Drag is just one term in this sum appearing as $-0.04 m |v_t| \vec{v}_t$. I suppose in the situation of your interest, the only other force is gravity. So you have $\vec F_{net}=-0.04 m |v_t| \vec{v}_t-mg\hat z=m \vec a$. This is a differential equation which you should solve to get the trajectory of the ball.

 

[mireazma]

Thanks for replying.
I need to find both A_t (acceleration) and v_t which, problematically, are interdependent: A depends on v as per given formula (A_t = v_t² * 0.04), v depends on A by Newton's laws. We ignore gravity. I don't know why you brought force into discussion.
I tried something on the paper but it yielded rubbish:
At v₀ = 120m/s it corresponds A₀ = 574m/s² (deceleration)
At a constant A it follows that after 0.1s it's supposed to have 62.6m/s. So now:
at v_0.1 = 62.6m/s A_0.1 = 156m/s².
But it's wrong as A continuously decreases as a function of v. During 0.1s the drag (A) has been dropping due to continuous deceleration, so v_0.1 is bigger than 62.6m/s.

 

[HallsofIvy]

That's why you need Calculus. Instantaneous acceleration is the derivative of velocity. You need to solve the "differential equation" $\frac{dv}{dt}= 0.04 v$ which can be separated as $\frac{1}{v} dv= 0.04 dt$. Integrate both sides.

 

[mireazma]

Thank you very much. It looks like what I'm asking for. Surprisingly as it may seem -- all the calculus I know is raw formulae for derivatives like (3x⁴)' = 12x³, sin(x)' = cos(x) etc. So this drastically outweighs my mathematical knowledge.
If I'm not asking too much, could you please, write an example so I can learn seeing the procedure?
You say some v₀ and a t, and obtain A_t and v_t. Please.

 

[vanhees71]

Let's solve the equation for the case of motion in one direction (it's more complicated for the general case). Then you have the differential equation
$$\dot{v}=-\alpha v^2,$$
with $\alpha$ a constant. This you can solve by "separation of variables":
$$\int_{v_0}^{v} \mathrm{d} v \frac{1}{v^2}=-\alpha \int_{0}^t \mathrm{d} t'=-\alpha t.$$
Integrating the left-hand side gives
$$-\alpha t=\frac{1}{v_0}-\frac{1}{v} \; \Rightarrow \; v(t)=\frac{1}{1/v_0+\alpha t}=\frac{v_0}{1+v_0 \alpha t}.$$

 

[mireazma]

Thank you very much for your time. Sorry for the late reply. Reading your post and "constant of integration" on Wikipedia raised some intriguing facts:
- A can be decomposed into terms of v and t making the formula simpler.
- v(t) can't be really determined as α isn't determined in the first place. But the problem, as it looks, must have an unique solution. It may be not determined mathematically? Maybe through practical measurements? And once I determine α from a measured v(t) I can calculate any v(t). But I can't measure, literally. And I can't think of any trivial cases to get some v(t) in order to determine α.
I tried using your formula with α = 1 as a neutral factor but for v(0) = 120 it gives v(0.1) = 9.23 which is not "it" for sure.

 

[vanhees71]

I don't understand what you mean; $\alpha$ is simply a constant, and I gave the full solution for the velocity.

 

[mireazma]

I'm a total rookie at calculus and this holds me back from getting to the bottom of many problems like this :( that's why I probably misunderstood your solution.
But to me it looks like this: α is an unknown constant -- constant yes, but yet unknown, undetermined; as such the recipe misses one ingredient. How do I determine α? Usually, I know that when there is an extra unknown, an extra equation is needed and sufficient. But I don't know how could I apply this to get α.
Am I missing something? I tried calculating v(0.1) and I couldn't.

I'm mentioning the problem is not for "academical" purposes but empirical. I'm into airsoft and I want to be able to calculate different bb velocities for different guns.