In this section we need to take a look at the velocity and
acceleration of a moving object.
From Calculus I we know that given the position function of
an object that the velocity of the object is the first derivative of the
position function and the acceleration of the object is the second derivative
of the position function.
So, given this it shouldn’t be too surprising that if the
position function of an object is given by the vector function then the velocity and acceleration of the
object is given by,
Notice that the velocity and acceleration are also going to
be vectors as well.
In the study of the motion of objects the acceleration is
often broken up into a tangential
component, aT, and a normal component, aN. The
tangential component is the part of the acceleration that is tangential to the
curve and the normal component is the part of the acceleration that is normal
(or orthogonal) to the curve. If we do
this we can write the acceleration as,
where and are the unit tangent and unit normal for the
If we define then the tangential and normal components of
the acceleration are given by,
where is the curvature
for the position function.
There are two formulas to use here for each component of the
acceleration and while the second formula may seem overly complicated it is
often the easier of the two. In the
tangential component, v, may be messy
and computing the derivative may be unpleasant. In the normal component we will already be
computing both of these quantities in order to get the curvature and so the
second formula in this case is definitely the easier of the two.
Let’s take a quick look at a couple of examples.
Example 1 If
the acceleration of an object is given by find the object's velocity and position
functions given that the initial velocity is and the initial position is .
We’ll first get the velocity. To do this all (well almost all) we need to
do is integrate the acceleration.
To completely get the velocity we will need to determine
the “constant” of integration. We can
use the initial velocity to get this.
The velocity of the object is then,
We will find the position function by integrating the
Using the initial position gives us,
So, the position function is,
Example 2 For
the object in the previous example determine the tangential and normal
components of the acceleration.
There really isn’t much to do here other than plug into
the formulas. To do this we’ll need to
Let’s first compute the dot product and cross product that
we’ll need for the formulas.
Next, we also need a couple of magnitudes.
The tangential component of the acceleration is then,
The normal component of the acceleration is,