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Vertical motion
The energy balance
From the values we have
determined
for the solid metal ball,
we may observe that the metal ball returns to us
with exactly the same speed of 20 m/s
which it had been given at the beginning of its motion.
Consequently, no kinetic energy has been lost during the trip
of the solid metal ball.
This is exactly what one may expect, as, without air resistance,
the only force which works on the solid metal ball
is the gravitational attraction.
During its trip upward the solid metal ball
recieves negative work
-mgymax
from the gravitational force mg,
since it works downward, opposite to the direction of motion.
For that reason the kinetic energy
which at the beginning of its trip was given to the solid metal ball,
diminishes during its upward voyage untill nothing is left
when the solid metal ball reaches its highest position
ymax.
On returning back to us,
the solid metal ball recieves positive work
mgymax
from the gravitational force mg,
since both, the direction of motion of the solid metal ball
and the gravitational attraction,
work in the same direction.
The net balance of work done by gravitation equals zero.
Hence the solid metal ball returns to us
with exactly the same amount of kinetic energy
as it had been given at the start of its trip.
That is the reason why the gravitational force is called
a conservative force.
A round trip has the effect of no net mechanical work
done by a conservative force,
exactly what we have observed.
more on the energy balance