Work and Energy
Energy Transformation for a
Dart
Consider an ordinary dart projection from a toy dart gun and
moving through the air. How could work and energy be utilized to
analyze the motion of the dart? Would the total mechanical energy of
the dart/gun system be altered when launched or while mvong through
the air? Or would the total mechanical energy of the dart/gun merely
be conserved?
Of course the answers to these questions begin by determining
whether or not there are any external forces doing work upon the
dart/gun system. According to the work-energy theorem, if external
forces do work upon the dart/gun system, the total mechanical energy
of the dart is not conserved; the initial amount of mechanical energy
is not the same as the final amount of mechanical energy. On
the other hand, if external forces do not do work upon the
dart/gun system, then the total mechanical energy is conserved; that
is, mechanical energy is merely transformed from one form to another
(say from potential to kinetic and/or vice versa) while the total
amount of the two forms remains unchanged.
In this case of the dart being launched from the spring gun, the
only forces doing work upon the dart are internal forces. Initially,
the dart is being acted upon by a spring force in order to be
projected from the dart gun. The coils of the springs are initially
compressed and upon pulling the trigger, the springs return to their
equilibrium position while pushing the dart out of the dart gun. The
dart then becomes a projectile (assuming their is negligible air
resistance); the only force doing work upon during its flight through
the air is gravity. Since both the spring force and the force of
gravity are internal forces, the total mechanical energy of the dart
is conserved. The animation below depicts the motion of the dart. The
animation is accompanied by work-energy bar charts which further
illustrate the tranformation of energy from one form to another and
the conservation of the total amount of mechanical energy.
The animation above shows that the energy of the dart/gun system
is initially present in the form of the elastic potential energy
(PEs) and gravitational potential energy (PEg).
The springs of the dart gun are compressed which accounts for the
elastic potential energy. Furthermore, the dart is initially elevated
at a height of 1-meter above the ground which accounts for the
gravitational potential energy. The presence of these two initial
forms of energy are shown by the PEg and PEs
bars of the bar chart. Once projected, the dart no longer has elastic
potential energy since the springs of the dart are no longer
compressed. However, the dart does have a large amount of kinetic
energy (energy of motion) since it is now moving at a high speed as
it leaves the dart. The dart also has gravitational potential energy
since it is still elevated to some height above the ground. As the
dart ascends towards its peak, it is continuously slowing down under
the influence of the downward force of gravity. During this ascent,
there is a transformation of the mechanical energy from the form of
kinetic energy (energy of motion) to gravitational potential energy
(the stored energy of vertical position). At the peak, there is only
a small amount of kinetic energy (the dart still has a horizontal
motion) and a large amount of gravitational potential energy (the
dart is at its highest vertical position). Finally, as the dart
descends to the ground, the force of gravity speeds it up. As it
falls, there is an increase in kinetic energy (due to the gain in
speed) and a decrease in gravitational potential energy (due to a
loss in vertical position).
The above animation provides a simple demonstration of how
mechanical energy merely changes form when the only forces acting
upon an object are internal forces. While the form of mechanical
energy changes, the total amount of mechanical energy (TME) is
conserved.
For more information on physical descriptions of motion, visit
The Physics Classroom Tutorial.
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