Written Materials

This document covers fundamental definitions of position, velocity, and acceleration that will be used throughout the course.

Video: The Position Vector

The position vector $$\vec r(t)$$ is used to express the location in space. In dynamics, that position vector will change in time. By defining that position vector, we can start describing that motion.

Video: The Velocity Vector

Velocity is a vector that is defined as the derivative of the position vector: $$\vec v (t_0) = \left. \frac{d \vec r}{dt} \right|_{t_0}$$. But what is the derivative of a vector? Well, we can use the definition:

$$\vec v (t_0) = \left. \frac{d \vec r}{dt} \right|_{t_0} = \lim_{\Delta t \rightarrow 0} \frac{\vec r(t_0 + \Delta t) \, – \, \vec r(t_0)}{\Delta t}.$$

The video below shows how to use the definition above to construct the velocity vector.

If you’re interested there is a similar video that lays out how to define the velocity vector when the object moves along a straight line.

The Acceleration Vector

Acceleration is a vector defined as the derivative of velocity.
$$\vec a (t_0) = \left. \frac{d \vec v}{dt} \right|_{t_0} = \lim_{\Delta t \rightarrow 0} \frac{\vec v(t_0 + \Delta t) \, – \, \vec v(t_0)}{\Delta t}.$$

It turns out that the nature of the acceleration vector is a bit different on straight and curvy paths, so we will consider both separately.

Video: Acceleration Along a Straight Line

We start with the simpler case. In the video, we construct the acceleration vector in the case that an object is moving back and forth along a straight line.

An Interactive Web Simulation (Spewton1D)

Click here to experiment with an interactive web simulation that illustrates the concepts above. [It is recommended that you use a desktop or laptop computer to run this simulation.]

Video: Acceleration Along Curved Paths

When an object moves along a curved path, it experiences a centripetal acceleration perpendicular to the path. The video shows where this acceleration comes from.

Elsewhere, I show you how to quantify this acceleration.

Spewton, A 2D Interactive Web Simulation

Click here to experiment with a 2D Interactive web simulation that illustrates the concepts above. It is recommended that you use a desktop or laptop computer to run this simulation.