Vector Calculus

Hokie dokie. Hopefully by this point you are familiar with parametric equations/vector equations, as we will be referring to them quite heavily throughout this wiki. If you are not familiar with them, get familiar now!!

Alright. Up to speed? Here we go. If you read the previous wiki on displacement, velocity, and acceleration, you understand that velocity is the derivative of displacement, acceleration is the derivative of velocity, and vice versa, ect., ect. This concept can be similarly applied to parametric equations, where your change in x and change in y represent the components of a vector describing your motion along a plane. Parametric equations can be expressed in the form x(t), y(t), or in the form r(t)=3i+4j, or similarly r(t)=x(t)i+y(t)j, where r(t) represents a vector equation for the position vector. Vector equations are basically the same as parametric equations because both are modeled by a seperate change of x and y.

Remembering what we know about displacement, velocity, and acceleration, it makes sense then that...

r'(t)=x'(t)+y'(t), with r'(t) being the velocity vector.



When you know what the velocity and acceleration vectors are at a point, you can then graph them and deduce information about the motion of the object from your vectors. When you graph velocity and acceleration vectors, the tail should start at the point being described by your velocity and acceleration vectors. In other words, graph your velocity and acceleration vectors at the head of your position vector.

Here's an example:
The position vector is represented by the black arrow, the velocity vector is represented by the green arrow pointing downwards and away from the graph, and the acceleration vector is on the concave side of the graph, pointing towards the y axis.

The direction of your velocity and acceleration vectors give information about how the object moving is being affected by these vectors. If the angle between the velocity and the acceleration vectors is acute, or less than 90 degrees, then the object is speeding up, because the acceleration vector is propelling the velocity vector. If angle between the velocity and acceleration vectors is greater than 90 degrees, the object is slowing down, because the acceleration vector is working against the velocity vector. If the angle is 90 degrees, then the particle is not speeding up or slowing down. However when looking at a graph showing the velocity and acceleration vectors, you can't always determine the exact angle between the two vectors. In this case, it's helpful to determine the angle mathematically.

When we want to determine the angle between two vectors, we go back to an equation you might remember from precalculus:

Now, when using this equation, you will make your life much easier if you can remember that cosx is positive for 0<x<90, cosx=0 at x=90, and cosx is negative for 90<x<180. Therefore, if cosx is positive, you know the object is speeding up because the angle between the two vectors is acute, and if cosx is negative, you know the object is slowing down because the angle between the two vectors is obtuse.

Here is an example running through the process of finding acceleration and velocity vectors, graphing them, and determining whether the object is speeding up or slowing down.

BUT WAIT, THERE'S MORE!!!! (How can you be so lucky?)

There are COMPONENTS of the acceleration vector--most importantly, the tangential component. The tangential component changes the speed of the object because it is the vector projection of the acceleration vector on the velocity vector. The tangential component is a vector in the direction of the velocity vector with magnitude |p|. You find p by following the equation:

P represents the scalar projection of the acceleration vector on the velocity vector. Additionally, the magnitude of the tangential component is the rate of change of the speed of the object. Furthermore:

Of course, there is another component of the acceleration vector: the normal component. This component changes the direction of the object's motion. Since we know that the acceleration vector is the sum of it's components, being the tangential and normal components, the normal component can be found by subtracting the tangential component from the entire acceleration vector.

Figure 10-6m taken from Calculus: Concepts and Applications by Foerster