# velocity and acceleration

**figure 1**:

**r**(

*t*), with

**v**(

*t*)
and

**a**(

*t*) shown.

Let's look at how position, velocity and acceleration are related and
can be written.

In the figure to the right, we show a space curve, which is
described by a position vector **r**(*t*). Then
**v**(*t*) gives the velocity at
any time *t*. The velocity, which is
**r**'(*t*), is also tangent to the
curve, as shown. The acceleration, which is **r**''(*t*) is
the same as
**v**'(*t*), which is the rate of change of velocity. So it
tells how fast the the velocity (change of position) is changing.

Ok, now let's think just about **v** and **a**. We saw in class
(and in our book) that **a** can be broken into exactly two pieces:
one part in the direction of **v**, which tells how much we're
speeding up or slowing down as we move along the space curve (that is,
that part of acceleration isn't trying to change the direction of
motion - just speed it up or slow it down), and the part perpendicular
to that, which is changing the actual direction of motion.

The first of these is the projection of **a** onto **v**:
**proj**_{v}**a** - as shown in
the lower diagram in the figure. And the perpendicular is what's left:
**a** - **proj**_{v}**a**.

So what? Ok, here's our main point:

We can break the acceleration of an object into exactly two pieces,
one in the direction of the velocity, which we call the

**tangential
component** of the acceleration, and the other perpendicular to
this, which we call the

**normal component**. With

**T**(

*t*), the unit tangent to the curve
( =

**r** '(

*t*)/|

**r** '(

*t*)| ) and

**N**(

*t*), the principle unit normal
( =

**T** '(

*t*)/|

**T** '(

*t*)| ), we can
therefore write

**a**(*t*) = *a*_{T}
**T**(*t*) + *a*_{N}
**N**(*t*).

And, not only that,

*a*_{T} is
just the length of the projection of

**a** onto

**v** (and

*a*_{N} **N**(

*t*)
can be found by subraction if we know

*a*_{T}).

**An Example**:

Let's see how this works in practice. Suppose that **r** =
<2*t*^{2}, *t*, 1>.
Find **v**, **a**, and *a*_{T} and *a*_{N}.

*Solution*:

The velocity and acceleration are easy to find:

**v** = **r**' = <4*t*, 1, 0>

**a** = **v**' = **r**'' = <2, 0, 0>.

Then, to find the tangential and normal components of acceleration we
need **T** and **N**, the *unit tangent* and
*principal unit normal* vectors. From their definitions, we
have

**T** = **v** / |**v**| =
<4*t*, 1, 0> /
sqrt(16*t*^{2} + 1),

and **N** = **T**' / |**T**'|, which isn't all
that nice to calculate. This is why we usually try and avoid
calculating it... Before doing that, notice that we can easily now
find *a*_{T}. This is just the
magnitude of the projection of **a** on **T** (see the
projections page):

*a*_{T} = |**a**| cos(theta)

= **a** . **T**,

because **a** . **T** =
|**a**| |**T**| cos(theta) and |**T**| = 1, so

*a*_{T} =
<2, 0, 0> .
<4*t*, 1, 0> /
sqrt(16*t*^{2} + 1)

= 8 *t* /
sqrt(16*t*^{2} + 1)

Then we can easily find
*a*_{N} **N** by subtracting:

**a** = *a*_{T} **T** +
*a*_{N} **N**, so

*a*_{N} **N** =
**a** - *a*_{T} **T**,
which is

*a*_{N} **N** =
<2, 0, 0> - ( 8 *t* /
sqrt(16*t*^{2} + 1) )
( <4*t*, 1, 0> /
sqrt(16*t*^{2} + 1) )

= <2, 0, 0> -
<32*t*^{2}, 8*t*, 0> /
(16*t*^{2} + 1).

Got all that? It's just a bunch of pushing vectors around. Go back
and make sure that it makes sense. We can carry out the subtraction, too:

*a*_{N} **N** =
<2, 0, 0> -
<32*t*^{2}, 8*t*, 0> /
(16*t*^{2} + 1)

=
<2, 0, 0> -
<32*t*^{2} /
(16*t*^{2} + 1),
8*t* / (16*t*^{2} + 1), 0>

=
< 2 - 32*t*^{2} /
(16*t*^{2} + 1),
-8*t* / (16*t*^{2} + 1), 0>

=
< 2 / (16*t*^{2} + 1),
-8*t* / (16*t*^{2} + 1), 0>

=
< 2, -8*t*, 0> /
(16*t*^{2} + 1)

Finally, if we wanted to find
*a*_{N} and **N**, the easiest
way to do it at this point would probably to find
*a*_{N}
by finding the magnitude of the vector
*a*_{N} **N** we found
above, and then find **N** by finding a unit vector in the direction
of the vector
*a*_{N} **N**.

But I'm not going to do that here. `:)`

velocity and acceleration

Last modified: Tue Sep 28 08:55:47 EDT 2004

Comments to:glarose@umich.edu

©2004 Gavin LaRose, UM Math Dept.