Simple Nature - Light and Matter

of gravity doesn’t change much if you only move up or down a few

meters. We also find that the gravitational energy is proportional

to the mass of the object we’re testing. Writingyfor the height,

andgfor the overall constant of proportionality, we have

Ug=mgy. [gravitational energy;y=height; only ac-

curate within a small range of heights]

The numberg, with units of joules per kilogram per meter, is called

thegravitational field. It tells us the strength of gravity in a certain

region of space. Near the surface of our planet, it has a value of

about 9.8 J/kg·m, which is conveniently close to 10 J/kg·m for

rough calculations.

Velocity at the bottom of a drop example 6

.If the skater in figure g drops 3 meters from rest, what is his

velocity at the bottom of the pool?

.Starting from conservation of energy, we have

0 =∆E

=∆K+∆U

=Kf−Ki+Uf−Ui

=

1

2

mvf^2 +mgyf−mgyi (becauseKi=0)

=

1

2

mvf^2 +mg∆y, (∆y<0)

so

v=

√

− 2 g∆y

=

√

−(2)(10 J/kg·m)(−3 m)

= 8 m/s (rounded to one sig. fig.)

There are a couple of important things to note about this ex-

ample. First, we were able to massage the equation so that it only

involved ∆y, rather thanyitself. In other words, we don’t need to

worry about wherey = 0 is; any coordinate system will work, as

long as the positiveyaxis points up, not down. This is no accident.

Gravitational energy can always be changed by adding a constant

onto it, with no effect on the final result, as long as you’re consistent

within a given problem.

The other interesting thing is that the mass canceled out: even

if the skater gained weight or strapped lead weights to himself, his

velocity at the bottom would still be 8 m/s. This isn’t an accident

either. This is the same conclusion we reached in section 1.2, based

on the equivalence of gravitational and inertial mass. The kinetic

energy depends on the inertial mass, while gravitational energy is

82 Chapter 2 Conservation of Energy