related to gravitational mass, but since these two masses are equal,
we were able to use a single symbol,m, for them, and cancel them
out.
We can see from the equationv=
√
− 2 g∆ythat a falling object’s
velocity isn’t constant. It increases as the object drops farther and
farther. What about its acceleration? If we assume that air friction
is negligible, the arguments in section 1.2 show that the accelera-
tion can’t depend on the object’s mass, so there isn’t much else the
accelerationcandepend on besidesg. In fact, the acceleration of a
falling object equals−g(in a coordinate system where the positive
yaxis points up), as we can easily show using the chain rule:
(
dv
dt)
=
(
dv
dK)(
dK
dU)(
dU
dy)(
dy
dt)
=
(
1
mv)
(−1)(mg)(v)=−g,where I’ve calculated dv/dKas 1/(dK/dv), and dK/dU=−1 can
be found by differentiatingK+U= (constant) to give dK+dU= 0.^7
We can also check that the units ofg, J/kg·m, are equivalent to
the units of acceleration,
J
kg·m=
kg·m^2 /s^2
kg·m=
m
s^2,
and therefore the strength of the gravitational field near the earth’s
surface can just as well be stated as 10 m/s^2.
Speed after a given time example 7
.An object falls from rest. How fast is it moving after two sec-
onds? Assume that the amount of energy converted to heat by
air friction is negligible.
.Under the stated assumption, we havea=−g, which can be
integrated to givev = −gt+ constant. If we lett = 0 be the
beginning of the fall, then the constant of integration is zero, so at
t= 2 s we havev=−gt=−(10 m/s^2 )×(2 s) = 20 m/s.
The Vomit Comet example 8
.The U.S. Air Force has an airplane, affectionately known as the
Vomit Comet, in which astronaut trainees can experience sim-
ulated weightlessness. Oversimplifying a little, imagine that the
plane climbs up high, and then drops straight down like a rock.
(It actually flies along a parabola.) Since the people are falling
with the same acceleration as the plane, the sensation is just like
what you’d experience if you went out of the earth’s gravitational(^7) There is a mathematical loophole in this argument that would allow the
object to hover for a while with zero velocity and zero acceleration. This point
is discussed on page 1022.
Section 2.1 Energy 83