4.2:
Average and Instantaneous Velocity Vectors
Average velocity is given as the ratio of displacement to the time interval during that displacement. In a two-dimensional space, the displacement vector, Δr, represents the change in position.
If the velocity is considered at any given moment, the elapsed time approaches zero. Now, the limit of the average velocity approaches instantaneous velocity and is equal to the derivative of the position vector with respect to time.
The direction of the instantaneous velocity vector at any point is always along a line tangent to the path at that point.
The components of velocity in a two-dimensional space are the time derivatives of the coordinates x and y. Thus, the instantaneous velocity vector is the sum involving two unit vectors.
When describing a 3-dimensional motion, the z-axis is also involved, and the velocity vector is represented using three unit vectors.
In the instantaneous velocity vector, the magnitude represents the speed of the object.
4.2:
Average and Instantaneous Velocity Vectors
To calculate other physical quantities in kinematics, the time variable must be introduced. The time variable not only allows us to state where an object is (its position) during its motion, but also how fast it’s moving. The speed at which an object is moving is given by the rate at which the position changes with time. For each position, a particular time is assigned. If the details of the motion at each instant are not important, the rate is usually expressed as the average velocity v. This velocity vector is simply the total displacement between two points divided by the time taken to travel between them, also known as the elapsed time.
However, since objects in the real world move continuously through space and time, the velocity of an object at any single point is an important parameter. The quantity that tells us how fast an object is moving anywhere along its path is known as the instantaneous velocity, often simply called velocity. It is the average velocity between two points on a path, in the limit where the time (and therefore the displacement) between the two points approaches zero. Like average velocity, instantaneous velocity is a vector with a dimension of length per time.
This text is adapted from Openstax, University Physics Volume 1, Section 4.2: Acceleration Vector.