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Vectors: Velocities,

Accelerations, and Forces

In order to understand the discoveries of Newton, we must have an
understanding of three basic quantities: (1) velocity, (2) acceleration,
and (3) force. In this section we define the first two, and in the next we
shall introduce forces. These three quantities have a common feature: they
are what mathematicians call *vectors*.

## Examples of Scalar Quantities

Vectors are quantities that require not only a magnitude, but a direction to
specify them completely. Let us illustrate by first citing some examples of
quantities that are not vectors. The number of gallons of gasoline in the fuel
tank of your car is an example of a quantitity that can be specified by a
single number---it makes no sense to talk about a "direction" associated with
the amount of gasoline in a tank. Such quantities, which can be specified by
giving a single number (in appropriate units), are called *scalars*.
Other examples of scalar quantities include the temperature, your weight,
or the population of a country; these are scalars because they are completely
defined by a single number (with appropriate units).
## Examples of Vector Quantities

However, consider a velocity. If we say that a car is going 70 km/hour, we
have not completely specified its motion, because we have not specified the
*direction* that it is going. Thus, velocity is an example of a vector
quantity. A vector generally requires more than one number to specify it; in
this example we could give the magnitude of the velocity (70km/hour), a
compass heading to specify the direction (say 30 degrees from North), and
an number giving the vertical angle with respect to the Earth's surface
(zero degrees except in chase scenes from action movies!).
The adjacent figure
shows a typical coordinate system for specifying a vector in terms of
a length *r* and two
angles, *theta* and *phi*.

## Graphical Representation of Vectors

Vectors are often distinguished from
scalar quantities either by placing a small arrow over the quantity, or by
writing the quantitity in a bold font. It is also common to indicate a vector
by drawing an arrow whose length is proportional to the magnitude of the
vector, and whose direction specifies the orientation of the vector.
In the adjacent image we show graphical representations for
three vectors. Vectors A and C have the same
magnitude but different directions. Vector B has the same orientation as
vector A, but has a magnitude that is twice as large. Each of these represents
a different vector, because for two vectors to be equivalent they must have
*both* the same magnitudes and the same orientations.

## Velocity and Acceleration

Let us now give a precise definition of velocity and acceleration. They are
vectors, so we must give a magnitude and a direction for them. The
velocity *v* and the acceleration
*a* are defined in the following
illustration,

This illustration also demonstrates graphically that velocity (and therefore
acceleration) is a vector: the
direction of the rock's velocity is certainly of critical interest to the
person standing under the rock in the two illustrations!
## Uniform Circular Motion is Accelerated Motion

Notice that velocity, which is a vector, is changed if either its magnitude or
its direction is changed. Thus, acceleration
occurs when either the magnitude or direction of the velocity (or
both) are altered.
In particular, notice from the adjacent image
that circular motion (even at uniform
angular velocity) implies a
continual acceleration, because the *direction* of the velocity
(indicated by the direction of the arrow) is
continuously changing, even if its magnitude (indicated by the length of the
arrow) is constant. This point, that
motion on a curved path is accelerated motion, will
prove crucial to our subsequent understanding of motion in gravitational
fields.

## How Many Accelerators Does Your Car Have?

Be aware that in popular speech acceleration is assumed to be an
*increase* in the magnitude of the velocity. As we have just seen,
acceleration also occurs when the direction of the velocity is changed, even if
the magnitude is constant; furthermore, in physics a *decrease* in the
velocity is just as much an acceleration as an increase.
Thus, your car actually has at least *3 accelerators*:
(1) the foot pedal called the "accelerator", that changes the magnitude of the
velocity, (2) the brake, which also changes the magnitude of the velocity, and
(3) the steering wheel, which changes the *direction* of the velocity!
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