# What exactly is “displacement” and “velocity”? [duplicate]

I'm finding two different definitions for what the displacement of an object is:

1) How far the object is from a starting point in a specified direction (direction represented through a straight line with an arrow).

2)The position vector of the point which represents the location of the object.

Aside from the fact that definition 1 gives us a scalar quantity and definition 2 gives us a vector quantity, these two definitions sound as if the application of them produce similar conceptual understanding, and that there's simply a difference in methodology.

However, this is not the case. When textbooks (and my teachers) use definition 1, this "specified direction" remains constant no matter what point in time we examine the position of the object. When we use definition 2 however, the direction can constantly change as time progresses. People using these different definitions will give different answers as to what the magnitude of displacement is at a given time.

Let me give you a thought experiment so you can see what I mean. Suppose we have an object moving in a two dimensional plane (with a y-axis and x-axis), where we measure the position of an object at different points in time through Cartesian Coordiantes.Let: "x=t" and "y=x^2", where "t" represents time elapsed in seconds.

Someone using definition 1 will choose a direction beforehand for making measurements. Let the starting point be (0,0). They might choose the "positive y-direction"(a vertical straight line with the arrow pointing upwards) for making measurements of displacement. After 3 seconds, the object will be at (3,9).Using the y coordinate we can see that (displacement)=9.

Now lets use definition 2 for examining the displacement after 3 seconds. The magnitude of displacement here is (90)^(0.5), by using the Pythagorean Theorem. (90)^(0.5)≠9. Also notice that the direction of this vector is obviously different from that in definition 1. By looking at different points in time we get different directions by applying definition 2.

And now for velocity. I'm fairly confident that velocity is the differential of displacement with respect to time, but because I don't know what the nature of displacement is, I don't know what the nature of velocity. Supposing that definition 2 is true, I would have to learn how differentiating works for vector quantities.

So I want to know what displacement REALLY is. Any answers would be much appreciated, since this topic is bugging me. This problem generalizes to understanding vector quantities as a whole. Anything mentioned about Forces would be a bonus for me, since I don't understand the proper way of describing them. I probably sound like a bit of an idiot entertaining definition 1 since displacement is described as a vector quantity from the get go (and not a scalar quantity) but I felt the need to do so in order to convey the frustration I'm feeling from reading conflicting sources of information.

## marked as duplicate by ZeroTheHero, M. Enns, sammy gerbil, Jon Custer, AccidentalFourierTransformApr 6 '18 at 13:50

Velocity can vary with time, it is given symbolically by:$\frac{dx}{dt}$