Game Development Reference

In-Depth Information

Particle Kinetics in 2D

As in particle kinematics, in particle kinetics you need to consider only the linear motion

of the particle. Thus, the equations of motion will consist of equations of the form
F
=

m
a
, where motion in each coordinate direction will have its own equation. The equa‐

tions for 2D particle motion are:

ΣF
x
= m a
x

ΣF
y
= m a
y

where ΣF
x
means the sum of all forces in the x-direction, ΣF
y
means the sum of all forces

in the y-direction, a
x
is the acceleration in the x-direction, and a
y
is the acceleration in

the y-direction.

The resultant force and acceleration vectors are:

a
= a
x
i
+ a
y
j

a
=
a
x
2
+
a
y
2

Σ
F
= ΣF
x
i
+ ΣF
y
j

ΣF =
(
ΣF
x
)
2
+
(
ΣF
y
)
2

Let's look at an example that appears simple but demonstrates the complexity of finding

closed-form solutions. A ship floating in water, initially at rest, starts up its propeller

generating a thrust,
T
, which starts the ship moving forward. Assume that the ship's

forward speed is slow and the resistance to its motion can be approximated by:

R = -C v

where
R
is the total resistance,
C
is a drag coefficient,
v
is the ship speed, and the minus

sign indicates that this resistive force opposes the forward motion of the ship. Find

formulas for the ship's speed, acceleration, and distance traveled as functions of time,

assuming that the propeller thrust and resistance force vectors act on a line of action

passing through the ship's center of gravity. This assumption lets you treat the ship as a

particle instead of a rigid body.

The first step in solving this problem is to identify all of the forces acting on the ship.

Figure 4-1
shows a
free-body diagram
of the ship with all of the forces acting on it—

namely, the propeller thrust,
T
; resistance,
R
; the ship's weight,
W
; and buoyancy,
B
.