For a computer game simulation, I want to compute velocity for a body whose acceleration magnitude is constant but whose acceleration direction may be rotating with a constant angular velocity.
I expect I have a basic misunderstanding or model problem. When I produce the velocity from the component-wise acceleration function, it seems that the speed of the direction change is inappropriately affecting the velocity calculation. I am hoping someone can tell me the basic problem with my model. I am not considering mass here so rotational momentum is not part of this model.
Let $a$ be the constant acceleration magnitude. Let $w$ be the constant angular velocity of change of acceleration direction and $d_0$ be the initial direction. So, considering just the x component, I think the acceleration function is $f_x(t) = a(cos(w t + d_0))$. If I take the antiderivative to be the velocity of the component, I get $$F_x(t) = \dfrac{a\sin\left(wt+d_0\right)}{w} + C$$.
My problem is with $w$ in the denominator. As $w$ approaches $0$ in the component acceleration function, the function approaches the (seemingly correct) $a\cos(d_0)$. But, as it approaches $0$ in the velocity function, the value becomes arbitrarily large.
With $w = 0$ I expect x-component velocity to be something like $(a\cos(d_0)) t + C$. But the velocity is undefined at $w = 0$ even though that produces a perfectly good acceleration value.
I wish to avoid a stepping/accumulation-based solution in my simulation, but instead have the ability to describe velocity (and ultimately position) as a function of time using this simplified dynamic model.
Your error is hidden in a lost (or wrongly used) integration constant. Indeed: $$ dv=a\cos(wt+\delta)\,dt\\ \implies\int_0^tdv=\int_0^ta\cos(wt+\delta)\,dt\\ \implies v-v_0=\left[\frac{a\sin(wt+\delta)}{w}\right]_0^t=\frac aw[\sin(wt+\delta)-\color{red}{\sin(\delta)}]. $$
The reason for the trouble with your solution is the missed term marked by red color. Indeed: $$ \lim_{w\to0}\frac aw[\sin(wt+\delta)-\sin(\delta)]=\lim_{w\to0}\frac aw[\sin wt\cos\delta-(1-\cos wt)\sin\delta]=at\cos\delta, $$ as it should be for constant acceleration.