Game Development Reference

In-Depth Information

where
f
is the weight,
m
is the mass, and
g
is the acceleration due to gravity. This

means that on different planets, the same object will have different weights (but the

same mass) because
g
changes.

On earth, we assume
g

=

10 m/s
2
, so an object with a weight of 1 kg will have a

weight of 1

10 kN. The kN unit is a unit of weight: kilograms, kg, are
not
a

unit of weight, despite what your bathroom scales might say! This causes scientists

who work on space prejects various problems: because
g
is different, they can no

longer convert English units such as pounds to kilograms using the conversion factors

found in science reference books. Pounds are a measure of weight, and kilograms are

a measure of mass.

So, back to buoyancy. Our block in the first part of figure 6.3 has a buoyancy force

of 10 kN. In the second part of the figure only half is submerged, so using the same

calculations, it has a buoyancy force of 5 kN.

Although we don't need to use it for our force generator, it is instructive to look

at the weight of the object too. In both cases the weight of the block is the same:

5 kN (a mass of 0.5 kg multiplied by the same value of
g
). So in the first part of

the figure, the buoyancy force will push the block upward; in the second part of the

figure the weight is exactly the same as the buoyancy, so the object will stay at the

same positionâ€”floating.

Calculating the exact buoyancy force for an object involves knowing exactly how

it is shaped because the shape affects the volume of water displaced, which is used to

calculate the force. Unless you are designing a physics engine specifically to model the

behavior of different-shaped boat hulls, it is unlikely that you will need this level of

detail.

Instead we can use a springlike calculation as an approximation. When the object

is near to the surface, we use a spring force to give it buoyancy. The force is pro-

portional to the depth of the object, just as the spring force is proportional to the

extension or compression of the spring. As we saw in figure 6.3, this will be accurate

for a rectangular block that is not completely submerged. For any other object it will

be slightly inaccurate, but not enough to be noticeable.

When the block is completely submerged, it behaves in a slightly different way.

Pushing it deeper into the water will not displace any more water, so as long as we

assume water has the same density, the force will be the same. The point-masses we

are dealing with in this part of the topic have no size, so we can't tell how big they

are to determine whether they are fully submerged. We can simply use a fixed depth

instead: when we create the buoyancy force, we give a depth at which the object is

considered to be fully submerged. At this point the buoyancy force will not increase

for deeper submersion.

By contrast, when the object is lifted out of the water, some part of it will still be

submerged until it reaches its maximum submersion depth above the surface. At this

point the last part of the object will have left the water. In this case there will be no

buoyancy force at all, no matter how high we lift the object: it simply is displacing no

more water.

Ă—

10

=