I would think yes because, if a rope tied to a swinging rock breaks, the rock flies off in the direction that is perpendicular to the direction of the last instant of the acceleration. The acceleration at the last instant it existed determined the velocity direction of the rock.
So acceleration over time changes a velocity’s speed and or direction but at any instant the velocity is determined by the acceleration at that instant.
Correct?
Instead of thinking of velocity and acceleration as two different things, simply think of acceleration as the name we've given to any change in velocity.
Then it suddenly is obvious that what we call acceleration is present in the very instance the velocity changes; not a moment before or a moment after. Because that change is acceleration, just in other words. Acceleration is not some odd "effect" that causes the change, it is the change.
We might still say it and describe it like that - we might still say that "acceleration causes the change in velocity" - because that's usually a neat, intuitive approach. But just remember that acceleration really is defined as the change itself.
The same logic holds true for force-vs-acceleration as well as other coupled properties.
"at any instant the velocity is determined by the acceleration at that instant."
No. The velocity at an instant, $t_1$, is determined by the accelerations at all previous times going back to a time $t_0$ at which we know the 'initial' velocity.
That's because we can use the accelerations to calculate the change in velocity, leading up to time $t_1$. Mathematically,
$$\vec v(t_1) = \vec v(t_0)\ + \int _{t_0} ^{t_1} \vec a(t) dt $$
Re. "at any instant the velocity is determined by the acceleration at that instant."
No. At a given point, the velocity and acceleration at an instant are independent of each other. Given acceleration at an instant, $t$, velocity at that instant can take on any value. However if acceleration $a'$ is given for an infinitesimal interval of time, $dt$, then the change in velocity during that time can be found as $a'dt$ assuming the acceleration is 'well behaved' so that it can be considered to be constant during the infinitesimal interval.
Note the difference between 'at an instant' and 'for an infinitesimal interval'. The interval allows to two quantities to be related through the derivative.
