In a two dimensional motion, instantaneous speed v₀ is a positive constant. Then which of the following are necessarily true?

The question asks about the characteristics of a two-dimensional motion where the instantaneous speed, denoted as $v_0$, is a positive constant.

Let us analyze the definition of instantaneous speed:
Instantaneous speed is defined as the magnitude of the instantaneous velocity vector:
$v=\left|\mathbf{v}\right|=\frac{ds}{dt}=v_0$
where $s$ represents the path length (or distance) traveled along the trajectory, and $t$ represents time.

Since $v_0$ is a constant, we can integrate the relation $ds=v_{0}dt$ over any time interval $\Delta t$:
$\Delta s=\int v_{0}dt=v_0\Delta t$
This shows that the path length (distance) traversed, $\Delta s$, in any time interval is directly proportional to the duration of the interval, $\Delta t$. Therefore, equal path lengths are traversed in equal intervals of time.

Let us evaluate why the other options are not necessarily true:
1. The average velocity is not zero at any time: If the object moves in a closed path (for example, uniform circular motion), the displacement after one complete revolution is zero, which means the average velocity for that time interval is zero. Thus, this is not necessarily true.
2. Average acceleration must always vanish: In circular motion with constant speed, there is a continuous change in the direction of velocity, resulting in a non-zero centripetal acceleration. Hence, the average acceleration does not necessarily vanish.
3. Displacements in equal time intervals are equal: Displacement depends on the direction of motion. If the direction of the velocity changes, the displacement vectors for equal time intervals will not be equal, even though the path lengths (scalar distances) are equal.

Thus, the only statement that is necessarily true is that equal path lengths are traversed in equal intervals.