Game Development Reference
In-Depth Information
The parameter
G
in Equation (3.5) is a constant value known as the
gravitational constant
.
It has a value of 6.674e-11
N-m
2
/kg
2
. The gravitational constant is referred to as a fundamental
physical constant because its value can be used to compute the gravitational force between
any two objects.
The distance
r
in Equation (3.5) is the distance between the centers of the two objects being
considered. The gravitational force acts along the line connecting the centers of the two objects,
meaning that the second object is pulled towards the center of the first object and vice versa.
When a skydiver jumps out of an airplane, she falls downwards because the gravitational force
due to the earth is pulling her towards the center of the earth. The magnitude of the gravitational
force depends on the masses of the objects involved. You exert a small gravitational force on
your car, but the earth exerts a much more significant gravitational force on you.
The magnitude of the gravitational force between two objects can be determined from
Equation (3.5). The acceleration of an object due to gravitational force can be calculated by
equating the gravitational force equation to Newton's second law.
Gm m
F
=
12
=
m a
(3.6)
G
2
1
r
In Equation (3.6),
m
1
is the mass of the object being considered and
a
is the acceleration of
the object. The quantity
m
1
is on both sides of the equation and cancels out.
Gm
a
=
2
(3.7)
2
r
We can see from Equation (3.7) that the acceleration of an object due to gravity is equal to
the gravitational constant multiplied by the mass of the other object divided by the square of the
distance between the centers of the objects. Because the mass of the object being considered
cancels out of the equation, we reach a very important conclusion.
The acceleration of an object
due to gravitational force is independent of the mass of the object
. If a bowling ball with a mass
of 5
kg
is dropped from an airplane, it will experience the same gravitational acceleration as
would a marble with a mass of 0.1
kg
.
■
Tidbit
Proving that the gravitational acceleration was constant irrespective of an object's mass was
shown rather dramatically by Galileo almost 400 years ago when he dropped cannon balls of different masses
from the Leaning Tower of Pisa. Sure enough, the cannon balls underwent the same acceleration and hit the
ground at the same time.
When incorporating gravitational effects into game programming applications, we will
usually be working with the acceleration due to Earth's gravity. The magnitude of Earth's grav-
itational acceleration can be computed from Equation (3.7). The mass of Earth is 5.9736
e
+24
kg
and the average radius of Earth is 6375
km
or 6.375
e
+6
m
.
m
s
6.674
e
−
11*5.9736
e
+
24
a
=
=
9.81
(3.8)
2
2
(
)
6.375
e
+
6
