The basic definition of work is
. This form of the work
allows us to obtain a quantity that has significant meaning in physics. That is,
starting from Newton's second law of motion,
, the work done by the force
along the displacement
In the last equality, the acceleration and
displacement are considered in the same direction; therefore, their scalar
product is the same as the algebraic product of their magnitudes (hyperlink
to general calculus). The expression for the work can be re-written using
relation for uniform accelerated motion,
. From where,
. Thus, the
In the previous expression, the effect of
work is to change the quantity
quantity is called the kinetic energy of the object moving at the velocity
Notices that the kinetic energy of an object is always a positive quantity
and independent of the direction of motion.
In addition, the kinetic energy of an object is not an
absolute property of the object. In fact, the kinetic energy of an object is
relative to the observer's frame of reference.
For example, in the drawing on the left, if observer
A throws a ball inside the bus with
corresponding kinetic energy of the ball, with respect to the people sitting in the
. Since observer B sees the same
ball moving at the velocity
is the velocity of
the bus with respect to observer B, he
will assign to the ball a kinetic energy given by
two energies are clearly different for each observer; being the kinetic energy
of the ball
measured by observer B greater than
the energy measured by observer A.
At the same time, the effect of ball hitting an object inside
the bus is clearly weaker than the effect of the ball hitting an object outside
the bus. Thus, this simple example illustrate that the kinetic energy is relative to the observer.
The unit of Kinetic Energy is the same as the unit of Work.
Unit of Kinetic Energy = (Kilogram)
(Meter/Second)2 = (Kilogram) (Meter/Second2) (Meter) =
Unit of Kinetic Energy = Joule
This result should be expected because the difference between
the final and initial kinetic energy is the work and only quantities of the same
nature can be added or subtracted.
The mathematical expression
, is called the
work-energy theorem because this formula establishes the relation between the
work and the kinetic energy,
This relation can be read in different forms,
If work is done by the external force in the
same direction as the displacement, the final
kinetic energy of the object is greater than
the original kinetic energy of the object.
if work is done by the external force in
opposite direction as the displacement, the
final kinetic energy of the object is smaller
than the original kinetic energy of the object.
If the kinetic energy of an object increases, work is
done on the object by an external force.
if the kinetic energy of an object decreases, the object
does work on the rest of the system. Therefore, the kinetic energy of an
object is the capacity of the object to perform work because of its motion.
If the net work is zero, the kinetic energy is unchanged.
This is the case of an object being lifted from the floor to a table when
the object starts at rest and with zero kinetic energy, and the object
ending with zero kinetic energy.
The change in kinetic energy is an observable that can be
measured experimentally for all classical objects; then, the change in kinetic
energy is the result of all forces acting on the object such as external agent
applying forces, frictional forces, normal force, or gravitational force. Thus,
The change in Kinetic Energy of a
system is equal to
the Net Work acting on the system even when the actual forces
acting on the system may be various or only part of one of the forces is the
responsible of the change (see application below).
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The graph on the left shows the necessary stopping
distance for a vehicle originally moving at different velocities when the
force applied by the brakes is constant and not reaction time (or
reaction distance) is considered in the analysis. In this case, the equations of
where the angle between the displacement of the car
and the external applied force (in this instance, the force of friction
with the road) is 1800. Moreover, at the end of the braking,
the car is at rest with the final velocity,
, equals to zero.
Therefore, the above equation is reduced to
For the graph on the left, the ratio between the mass and the force is
Notices that the independent coordinate (horizontal) shown in the graph is the
velocity while the dependent variable (vertical) is the displacement.
Analyzing the relation,
it can be observed that the stopping distance may depend on the mass; that is,
an increment on the mass of the car produce a linear increment on the
stopping distance of the car. However, the
frictional force also depend
on the mass of the object,
is the coefficient of friction between the surfaces. If it is an object
sliding over another surface, it is the coefficient of kinetic friction.
On the other side, if it is a wheel turning over a surface, it is the
coefficient of static friction assuming that the maximum braking occurs
when the static friction is maximum. In either case, the stopping