# Notes on Vectors and Centripetal Acceleration

```VECTORS AND CENTRIPETAL ACCELERATION
We have just seen that position, displacement, velocity
and acceleration each have associated direction, as well
as a magnitude. In general, quantities which have both a
numerical value and a direction are called vectors.
By contrast, quantities which have only numerical value
(i.e. no direction) are called scalars. In our previous
discussion, distance and speed are scalars. (Other well
known scalars are mass, time, and temperature.)
For the purpose of visualizing vectors, we often draw
them using arrows, with the direction of the arrow
(obviously!) showing the direction of the vector, while
the length of the arrow conveys its magnitude. (For
example, look at the various velocity vectors on p 189 of
the textbook). We can also use a positive numerical
factor to expand or shrink a particular vector (in this
case called V), and a minus sign to reverse its direction,
as the following illustrations show.
V
2V
2
V
3
V
3
 V
2
There is one other important point regarding vectors to
be made here. A vector is completely determined by its
magnitude (or length) and its direction. In particular, in
2
most applications, the location of a vector is immaterial,
i.e. moving a vector to a different location does not
change the vector as long as the magnitude and the
direction remain unchanged.
This point is illustrated in Figures i, ii and iii below.
U
U
U
V
V
i
ii
V
iii
Note that V has been positioned in three different
locations relative to U (completely separated, “tail-totail”, “tail-to-tip”), but since their length and direction
have not been changed, the vector is still V in each case!
You won’t have to draw vectors often (if at all) in this
course, but it can be useful to visualize vectors
occasionally, for the purpose of explanation. Most of the
time, you will recognize a vector not by its
representation with an arrow, but simply because it is
clearly given an associated direction (such as being told
that objects have velocities of 5 m/s east or 80 km/h
south-west).
We will use vector diagrams again in the next discussion,
but only to help explain the concept of centripetal
acceleration.
3
CENTRIPETAL ACCELERATION
Consider a body traveling counterclockwise around a
circular path at constant speed, and suppose we look at
its position and velocity vectors at two times which are
separated by a relatively small time interval t.
The initial and final position and velocity vectors
(denoted ri, rf, vi and vf) are shown in the following
diagram.
vi
vf
rf
ri
The position vectors ri and rf are certainly different
(since their directions aren’t the same) but, since the
path is a circle, they have the same length (which we will
call simply r). Likewise, the velocity vectors vi and vf
point in different directions, but, since the speed is
constant, they also have the same length (which we will
call simply v).
4
Because vectors can be moved without changing them
(as long as the length and direction remain the same, we
can produce the following new diagrams:
r
vi
v

vf
rf
ri

Note that we have added the vectors representing the
displacement (or change in position) r and the change
in velocity v. It is easy to see that these two triangles
are similar triangles, and hence that their corresponding
sides have a constant ratio (remember that the concept
of similar triangles was an important historical milestone
of science!). As a result, we can write (since the lengths
of vi and ri are just v and r):
v
r
v r


or,
dividing
by
t,
v
r
v t r  t .
5
But, according to the definitions of acceleration and
velocity,
v
r
 a and
v.
t
t
This means that
a v
v2
 , or, multiplying by v, a 
v r
r .
Finally, note in the above diagram that the direction of
v (and hence the direction of the acceleration a) is
always perpendicular to the direction of the
displacement r.
So, to summarize, when an object travels around a circle
of radius r with constant speed v, it experiences an
acceleration
v2
a
r
which is directed toward the center of the circle.
Because it is “center-pointing”, this special type of
acceleration has been given the name centripetal
acceleration. Remember, again, that the speed is
constant, showing that a body can accelerate without
changing speed! This is because the velocity vector is
changing direction even though its magnitude stays the
same.
6
This means that centripetal acceleration can always be
calculated for any object traveling in a circle, if we
know: (i) the radius of the circle; and (ii) the speed of the
object. For example, a person sprinting around a circle
of radius 9 m with speed 6 m/s is accelerating toward the
center of the circle with
v 2 62
a

 4 m/s2.
r
9
The notion of centripetal acceleration, and this formula
in particular, will be very important later on in our
discussion of gravity.
```