can velocity and acceleration vectors ever be perpendicular? Parallel?

Answer 1

yes. For both

velocity and acceleration are parallel when an object travels in a straight line


Velocity and acceleration are perpendicular to each other when an object travels in a circle

Answer 2

Sure they can. For example, you are on a ferris wheel. As the ferris wheel goes around, there is centripetal force and there is centrifugal force (sort of, but that's a different story). These forces act on you in the direction of the spokes of the ferris wheel and they are there whenever the wheel is rotating.

Wherever there is force, there is also acceleration from f = ma; where f is the force, m is your mass, and a is that centripetal or centrifugal acceleration along the spokes of the wheel. Thus, there is acceleration acting along the direction of the spokes in the ferris wheel.

But in addition to those two forces and their spoke oriented acceleratations, you are also going around the hub of the wheel. So there is tangential velocity on the rim of the ferris wheel, where you are sitting. Tangential velocity is perpendicular to the spokes of the turning wheel.

So there you have, on that ferris wheel, your acceleration going along the spokes while your tangential velocity is going perpendicular to that acceleration.

Can you have two accelerations perpendicular? Sure, but as acceleration is a vector, two accelerations going perpendicular would add up to a single acceleration where A^2 = a1^2 + a2^2; and A is the rate of the resulting acceleration from a1 and a2 acting perpendicular to each other.

Same thing for velocities, which are also vectors. That is, two perpendicular velocities add up to a resulting speed (the magnitude of the velocities) as V^2 = v1^2 + v2^2.

A common place we find perpendicular accelerations and velocities is when we have x and y directions; so we separate a single acceleration A into ax and ay, the accelerations in the x and y direction. Or we write V, a single velocity, as vx and vy, the perpendicular components of V.

Parallel accelerations and velocities would occur if and only if they were accelerations and velocities of different bodies accelerating and moving. This results because "parallel" means going in the same direction (or opposite) so they will never cross each other, but separated from each other. If they were not separated by some space, they would be coincident, not parallel, if they were going in the same or dierectly opposite direction.