Direction of velocity and displacement Do velocity and displacement always have the same direction? I know this seems like a silly question to ask, but I'm having a hard time coming up with any counter examples, and I'm not able to prove the statement either.
I can think of plenty of examples where acceleration has a different direction than that of velocity (like circular motion, retardation of a car etc), but I'm unable to do so for velocity and displacement. Can someone help me out with this?
 A: The velocity vector describes the change in the displacement vector, and hence, it can be in any direction relative to what direction the displacement vector points in, since there are many ways the position of the particle can change.
A few counterexamples:

*

*Circular motion: Velocity vector is perpendicular to the position vector which denotes the particle on path

*Harmonic motion: at extreme ends of the motion, the velocity vector has opposite sign to that of displacement vector.

A: Displacement is the shortest possible distance between the initial point and final point of line of motion.

Do velocity and displacement always have the same direction?

Need not to be.
Imagine a ball being projected vertically upward with some speed $v$. At some time t (less than time of ascent i.e. $t<\frac{v}{g}$ ), the direction of its velocity is upward  and the direction of the displacement of the ball as seen from from its point of projection is also upward (and the magnitude of displacement is increasing with time) . So both have same direction.

But after the ball reaches the maximum height and starts to come back ,
what's the direction of its velocity in the mid way ?
The ball is coming down, so the direction of velocity is downward and now what is the direction of displacement of the body from the point of projection ?
It is still in upward direction (but this time the magnitude of displacement is decreasing).

So you got that. !!
NOTE : $h$  and $H_1$ in the two pictures represent the
magnitude and the arrows show the direction of displacement respectively . I is used to indicate the initial point of motion.
Hope it helps ☺️.
